Chapter 11
Boundary-Value Problems for Ordinary Differential Equations
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After completing this chapter, the reader should be able to:
formulate linear and nonlinear two-point boundary-value problems;
distinguish initial-value problems from boundary-value problems;
implement shooting and multiple-shooting methods;
derive finite-difference discretizations for linear and nonlinear BVPs;
solve nonlinear finite-difference systems by Newton’s method;
formulate collocation methods, including polynomial and Chebyshev collocation;
derive weak formulations for self-adjoint linear BVPs;
build Galerkin approximations for linear BVPs;
build Newton–Galerkin methods for nonlinear BVPs;
assemble finite-element stiffness matrices and load vectors;
evaluate Galerkin integrals using high-order Gauss–Legendre quadrature;
understand adaptive refinement, residual estimators, and practical solver choices.
Boundary-Value Problems
A second-order two-point boundary-value problem has the form y''=f(x,y,y'), \qquad a<x<b, with boundary conditions such as y(a)=\alpha, \qquad y(b)=\beta. These are Dirichlet boundary conditions. More general boundary conditions include Neumann conditions y'(a)=\gamma, \qquad y'(b)=\delta, and Robin conditions \eta_a y(a)+\mu_a y'(a)=\gamma_a, \qquad \eta_b y(b)+\mu_b y'(b)=\gamma_b.
An initial-value problem evolves from complete data at one point. A boundary-value problem imposes conditions at two or more points. This changes the numerical structure: BVP discretizations usually produce algebraic systems, often nonlinear.
A standard linear model problem is -y''(x)+q(x)y(x)=g(x), \qquad y(a)=\alpha,\quad y(b)=\beta. A self-adjoint variable-coefficient model is -\frac{\dd}{\dd x}\left(p(x)y'(x)\right)+q(x)y(x)=g(x), \qquad y(a)=\alpha,\quad y(b)=\beta, where p(x)\ge p_0>0, \qquad q(x)\ge0.
Well-Posedness and Maximum Principles
For linear BVPs, numerical methods are meaningful only when the continuous problem is well posed.
Suppose p(x)\ge p_0>0, \qquad q(x)\ge0 on [a,b]. Then the homogeneous problem -\frac{\dd}{\dd x}\left(p(x)y'(x)\right)+q(x)y(x)=0, \qquad y(a)=0,\quad y(b)=0, has only the trivial solution.
Multiply the equation by y and integrate: \int_a^b \left[ -\left(p y'\right)'y+q y^2 \right]\dd x=0. Integration by parts gives \int_a^b p(x)|y'(x)|^2\,\dd x + \int_a^b q(x)|y(x)|^2\,\dd x = \left[-p(x)y'(x)y(x)\right]_a^b. The boundary term is zero because y(a)=y(b)=0. Hence \int_a^b p|y'|^2\,\dd x+\int_a^b q|y|^2\,\dd x=0. Both integrals are nonnegative. Since p\ge p_0>0, we obtain y'=0 almost everywhere. Thus y is constant. The boundary condition gives y=0.
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Shooting Method
Consider y''=f(x,y,y'), \qquad y(a)=\alpha, \qquad y(b)=\beta. Introduce u_1=y, \qquad u_2=y'. Then u_1'=u_2, \qquad u_2'=f(x,u_1,u_2). The missing initial slope is s=y'(a). For each s, solve the IVP u_1(a)=\alpha, \qquad u_2(a)=s. Define the shooting residual F(s)=u_1(b;s)-\beta. The BVP is solved by finding F(s)=0.
Caption.
Single shooting method
BVP y''=f(x,y,y'), boundary values \alpha,\beta, initial slope guess s_0
For k=0,1,2,\ldots:
Solve the IVP with y(a)=\alpha, y'(a)=s_k
Compute F(s_k)=y(b;s_k)-\beta
If |F(s_k)| is small enough:
Return y(x;s_k)
Update s_k using Newton, secant, or bisection
For the nonlinear BVP y''=f(x,y,y'), let y(x;s) be the shooting solution with y(a;s)=\alpha, \qquad y'(a;s)=s. The derivative z(x;s)=\frac{\partial y(x;s)}{\partial s} satisfies z''=f_y(x,y,y')z+f_{y'}(x,y,y')z', \qquad z(a)=0, \qquad z'(a)=1. Therefore F'(s)=z(b;s).
Differentiate y''(x;s)=f(x,y(x;s),y'(x;s)) with respect to s. This gives z''=f_yz+f_{y'}z'. Differentiating the initial conditions y(a;s)=\alpha, \qquad y'(a;s)=s with respect to s yields z(a)=0, \qquad z'(a)=1. Since F(s)=y(b;s)-\beta, we have F'(s)=z(b;s).
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Shooting may be ill-conditioned when the associated IVP has exponentially growing modes. Small changes in the initial slope can create huge changes at the right boundary. Multiple shooting and collocation are often more robust.
Multiple Shooting
Multiple shooting divides [a,b] into subintervals a=x_0<x_1<\cdots<x_M=b. On each subinterval, an IVP is solved with unknown initial data. Continuity conditions are imposed at the internal nodes: y_j(x_j)=Y_j, y_j(x_{j+1})=Y_{j+1}. The unknowns are the interface values and possibly slopes. The resulting nonlinear system is solved by Newton’s method.
Multiple shooting reduces sensitivity because each IVP is solved on a shorter interval.
Finite Differences for Linear BVPs
Consider -y''(x)=g(x), \qquad y(a)=\alpha, \qquad y(b)=\beta. Let x_i=a+ih, \qquad i=0,\ldots,N, \qquad h=\frac{b-a}{N}. For interior points, y''(x_i) \approx \frac{y_{i-1}-2y_i+y_{i+1}}{h^2}. Thus -\frac{y_{i-1}-2y_i+y_{i+1}}{h^2}=g(x_i), \qquad i=1,\ldots,N-1. This produces a tridiagonal linear system.
If y\in C^4([a,b]), then \frac{y(x_i-h)-2y(x_i)+y(x_i+h)}{h^2} = y''(x_i)+O(h^2).
Taylor expansions give y(x_i+h)=y(x_i)+hy'(x_i)+\frac{h^2}{2}y''(x_i) +\frac{h^3}{6}y^{(3)}(x_i)+\frac{h^4}{24}y^{(4)}(\xi_+), and y(x_i-h)=y(x_i)-hy'(x_i)+\frac{h^2}{2}y''(x_i) -\frac{h^3}{6}y^{(3)}(x_i)+\frac{h^4}{24}y^{(4)}(\xi_-). Adding and rearranging yields \frac{y(x_i-h)-2y(x_i)+y(x_i+h)}{h^2} = y''(x_i)+O(h^2).
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Caption.
Finite-difference method for -y''=g
interval [a,b], boundary values \alpha,\beta, number of panels N
h\gets(b-a)/N
For i=1,\ldots,N-1:
Set row i of the tridiagonal system:
\item -Y_{i-1}+2Y_i-Y_{i+1}=h^2 g(x_i) \item
Move known boundary terms Y_0=\alpha, Y_N=\beta to the right-hand side
Solve the tridiagonal system for Y_1,\ldots,Y_{N-1}
Return Y_0,\ldots,Y_N
Finite Differences for Nonlinear BVPs
For y''=f(x,y,y'), \qquad y(a)=\alpha, \qquad y(b)=\beta, use y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}, and y'(x_i)\approx \frac{Y_{i+1}-Y_{i-1}}{2h}. The nonlinear equations are R_i(Y)= \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2} - f\left( x_i,Y_i,\frac{Y_{i+1}-Y_{i-1}}{2h} \right) = 0.
Newton’s method solves J(Y^{(k)})\Delta Y^{(k)} = -R(Y^{(k)}), then updates Y^{(k+1)}=Y^{(k)}+\Delta Y^{(k)}.
Caption.
Newton finite-difference method for nonlinear BVPs
mesh x_i, boundary values, initial guess Y^{(0)}
For k=0,1,2,\ldots:
Assemble nonlinear residual R(Y^{(k)})
Assemble Jacobian J(Y^{(k)})
Solve J(Y^{(k)})\Delta Y^{(k)}=-R(Y^{(k)})
Y^{(k+1)}\gets Y^{(k)}+\Delta Y^{(k)}
If \|R(Y^{(k+1)})\| and \|\Delta Y^{(k)}\| are small:
Return Y^{(k+1)}
Collocation Methods
Collocation methods approximate the solution by a polynomial or piecewise polynomial u_h and enforce the differential equation at selected points: \mathcal L u_h(x_j)=g(x_j). For a nonlinear operator, \mathcal N(u_h)(x_j)=0.
For example, on [a,b], choose a polynomial u_N(x)=\sum_{k=0}^{N} c_k\phi_k(x) and impose u_N(a)=\alpha, \qquad u_N(b)=\beta, and u_N''(\xi_j)=f(\xi_j,u_N(\xi_j),u_N'(\xi_j)), \qquad j=1,\ldots,N-1.
Chebyshev Spectral Collocation
Map [a,b] to [-1,1] by x=\frac{b-a}{2}\xi+\frac{a+b}{2}. Use Chebyshev–Gauss–Lobatto nodes \xi_j=\cos\left(\frac{\pi j}{N}\right), \qquad j=0,\ldots,N. Let D_\xi be the Chebyshev differentiation matrix on [-1,1]. Then D_x=\frac{2}{b-a}D_\xi, \qquad D_x^{(2)}=\left(\frac{2}{b-a}\right)^2D_\xi^2. For the linear problem -y''+q(x)y=g(x), \qquad y(a)=\alpha, \qquad y(b)=\beta, the interior collocation equations are -\left(D_x^{(2)}Y\right)_j+q(x_j)Y_j=g(x_j), \qquad j=1,\ldots,N-1, with boundary equations Y(a)=\alpha, \qquad Y(b)=\beta.
If the solution of a linear BVP is analytic in a complex neighborhood of [a,b] and the boundary-value problem is well conditioned, then Chebyshev collocation can converge faster than any algebraic power of N^{-1}.
Analyticity implies geometric decay of Chebyshev coefficients. The Chebyshev interpolant and its differentiated approximations converge rapidly in appropriate norms. Since the differential operator is imposed at the collocation nodes and the continuous BVP is assumed well conditioned, a small collocation residual implies a small solution error. Therefore the convergence is spectral until limited by roundoff or conditioning.
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Weak Formulation for Linear BVPs
Consider the homogeneous Dirichlet problem -\frac{\dd}{\dd x}\left(p(x)y'(x)\right)+q(x)y(x)=g(x), \qquad y(a)=0,\quad y(b)=0. Let V=H_0^1(a,b). Multiply by a test function v\in V and integrate: \int_a^b \left[ -\left(p y'\right)'v+qyv \right]\dd x = \int_a^b gv\,\dd x. Integrating by parts gives the weak formulation: \int_a^b p y'v'\,\dd x + \int_a^b q yv\,\dd x = \int_a^b gv\,\dd x \qquad \forall v\in V. Define a(y,v)= \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x, and \ell(v)=\int_a^b gv\,\dd x. The weak problem is: \text{Find }y\in V \text{ such that } a(y,v)=\ell(v) \quad \forall v\in V.
If p(x)\ge p_0>0 and q(x)\ge0, then a(v,v)\ge c\|v\|_{H^1(a,b)}^2 \qquad \forall v\in H_0^1(a,b), for some c>0.
We have a(v,v)=\int_a^b p|v'|^2\,\dd x+\int_a^b q|v|^2\,\dd x \ge p_0\|v'\|_{L^2}^2. By Poincare’s inequality on H_0^1(a,b), \|v\|_{L^2}\le C_P\|v'\|_{L^2}. Hence \|v\|_{H^1}^2 = \|v\|_{L^2}^2+\|v'\|_{L^2}^2 \le (C_P^2+1)\|v'\|_{L^2}^2. Therefore a(v,v)\ge \frac{p_0}{C_P^2+1}\|v\|_{H^1}^2.
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Linear Galerkin Method
Let V_h\subset V be a finite-dimensional trial and test space with basis \phi_1,\ldots,\phi_m. The Galerkin approximation is: \text{Find }y_h\in V_h \text{ such that } a(y_h,v_h)=\ell(v_h) \quad \forall v_h\in V_h. Write y_h=\sum_{j=1}^{m}Y_j\phi_j. Testing with v_h=\phi_i gives \sum_{j=1}^{m} a(\phi_j,\phi_i)Y_j = \ell(\phi_i), \qquad i=1,\ldots,m. Thus KY=F, where K_{ij}=a(\phi_j,\phi_i) = \int_a^b p\phi_j'\phi_i'\,\dd x + \int_a^b q\phi_j\phi_i\,\dd x, and F_i=\int_a^b g\phi_i\,\dd x.
Let y\in V solve the weak problem and y_h\in V_h solve the Galerkin problem. Then a(y-y_h,v_h)=0 \qquad \forall v_h\in V_h.
The exact solution satisfies a(y,v_h)=\ell(v_h) \qquad \forall v_h\in V_h. The Galerkin solution satisfies a(y_h,v_h)=\ell(v_h) \qquad \forall v_h\in V_h. Subtracting gives a(y-y_h,v_h)=0.
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Assume a(\cdot,\cdot) is continuous and coercive on V: |a(u,v)|\le M\|u\|_V\|v\|_V, \qquad a(v,v)\ge \alpha\|v\|_V^2. Then \|y-y_h\|_V \le \frac{M}{\alpha} \inf_{v_h\in V_h}\|y-v_h\|_V.
For any v_h\in V_h, coercivity gives \alpha\|y-y_h\|_V^2 \le a(y-y_h,y-y_h). Using Galerkin orthogonality with y_h-v_h\in V_h, a(y-y_h,y_h-v_h)=0. Therefore a(y-y_h,y-y_h)=a(y-y_h,y-v_h). By continuity, a(y-y_h,y-v_h) \le M\|y-y_h\|_V\|y-v_h\|_V. Canceling \|y-y_h\|_V and taking the infimum over v_h gives the result.
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Finite Elements with Piecewise Linear Basis Functions
Let a=x_0<x_1<\cdots<x_N=b. The piecewise linear hat basis functions satisfy \phi_i(x_j)=\delta_{ij}. For homogeneous Dirichlet conditions, the unknown basis functions are \phi_1,\ldots,\phi_{N-1}.
On an element e=[x_i,x_{i+1}], with length h_e=x_{i+1}-x_i, the local basis functions are \varphi_1(x)=\frac{x_{i+1}-x}{h_e}, \qquad \varphi_2(x)=\frac{x-x_i}{h_e}. Their derivatives are \varphi_1'(x)=-\frac1{h_e}, \qquad \varphi_2'(x)=\frac1{h_e}. For constant p, the local stiffness matrix is K^{(e)} = \frac{p}{h_e} \begin{pmatrix} 1&-1\\ -1&1 \end{pmatrix}. For constant q, the local mass matrix is M^{(e)} = \frac{q h_e}{6} \begin{pmatrix} 2&1\\ 1&2 \end{pmatrix}.
Caption.
Finite-element assembly for a linear BVP
mesh x_0,\ldots,x_N, coefficients p,q, source g
Initialize global matrix K and vector F to zero
For each element e=[x_i,x_{i+1}]:
Compute local stiffness entries
\item K_{mn}^{(e)}=\int_{x_i}^{x_{i+1}} \item \left(p\varphi_n'\varphi_m'+q\varphi_n\varphi_m\right)\dd x \item
Compute local load entries
\item F_m^{(e)}=\int_{x_i}^{x_{i+1}}g\varphi_m\,\dd x \item
Add local entries to global matrix and vector
Apply boundary conditions
Solve the resulting linear system
High-Precision Quadrature for Galerkin Integrals
Galerkin and finite-element methods require integrals such as \int_{x_i}^{x_{i+1}} p(x)\varphi_m'(x)\varphi_n'(x)\,\dd x, \int_{x_i}^{x_{i+1}} q(x)\varphi_m(x)\varphi_n(x)\,\dd x, and \int_{x_i}^{x_{i+1}} g(x)\varphi_m(x)\,\dd x. If the coefficients are not constant, these integrals must be approximated accurately.
Use the reference element \xi\in[-1,1], \qquad x=x_c+\frac{h_e}{2}\xi, \qquad x_c=\frac{x_i+x_{i+1}}{2}. Then \int_{x_i}^{x_{i+1}} F(x)\,\dd x = \frac{h_e}{2} \int_{-1}^{1} F\left(x_c+\frac{h_e}{2}\xi\right)\dd \xi. An r-point Gauss–Legendre rule is \int_{-1}^{1}F(\xi)\,\dd \xi \approx \sum_{\ell=1}^{r}w_\ell F(\xi_\ell), ] and is exact for all polynomials of degree at most \[ 2r-1.
Explicit low-order Gauss–Legendre rules are: r=1: \qquad \xi_1=0, \qquad w_1=2. r=2: \qquad \xi_{1,2}=\pm\frac1{\sqrt3}, \qquad w_1=w_2=1. r=3: \qquad \xi_1=0,\quad \xi_{2,3}=\pm\sqrt{\frac35}, w_1=\frac89, \qquad w_2=w_3=\frac59. r=4: \qquad \xi_{1,2}=\pm \sqrt{\frac{3}{7}-\frac{2}{7}\sqrt{\frac65}}, \qquad \xi_{3,4}=\pm \sqrt{\frac{3}{7}+\frac{2}{7}\sqrt{\frac65}}, w_{1,2}= \frac{18+\sqrt{30}}{36}, \qquad w_{3,4}= \frac{18-\sqrt{30}}{36}. r=5: \qquad \xi_1=0, \xi_{2,3}=\pm\frac13\sqrt{5-2\sqrt{\frac{10}{7}}}, \qquad \xi_{4,5}=\pm\frac13\sqrt{5+2\sqrt{\frac{10}{7}}}, w_1=\frac{128}{225}, w_{2,3}=\frac{322+13\sqrt{70}}{900}, \qquad w_{4,5}=\frac{322-13\sqrt{70}}{900}.
An r-point Gauss–Legendre rule on [-1,1] integrates exactly every polynomial of degree at most 2r-1.
The Gauss nodes are the roots of the Legendre polynomial P_r, which is orthogonal to all polynomials of degree at most r-1. For any polynomial p of degree at most 2r-1, divide by P_r: p=P_r q+s, \qquad \deg q\le r-1, \qquad \deg s\le r-1. The quadrature rule is interpolatory on r nodes, so it integrates s exactly. At the nodes, P_r=0, so the quadrature of P_rq is zero. The exact integral of P_rq is also zero by orthogonality. Therefore the rule integrates p exactly.
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For piecewise polynomials of degree k: \varphi_i\varphi_j \quad \text{has degree }2k, and \varphi_i'\varphi_j' \quad \text{has degree }2k-2. If coefficients p,q,g are polynomial, choose Gauss order r so that 2r-1 is at least the degree of the full integrand. If coefficients are smooth but not polynomial, increase r until the quadrature error is negligible compared with the discretization error.
Caption.
High-order Gauss quadrature on an element
element [x_i,x_{i+1}], integrand F(x), Gauss nodes \xi_\ell, weights w_\ell
h_e\gets x_{i+1}-x_i, x_c\gets(x_i+x_{i+1})/2
I\gets0
For \ell=1,\ldots,r:
x_\ell\gets x_c+(h_e/2)\xi_\ell
I\gets I+w_\ell F(x_\ell)
Return (h_e/2)I
Nonlinear Galerkin Method
Consider a nonlinear BVP written abstractly as \mathcal N(y)=0 with boundary conditions. A typical example is -y''+q(x)y+\gamma y^3=g(x), \qquad y(a)=0,\quad y(b)=0. Multiplying by v\in V and integrating gives: \int_a^b y'v'\,\dd x + \int_a^b q yv\,\dd x + \int_a^b \gamma y^3v\,\dd x - \int_a^b gv\,\dd x = 0. Define the nonlinear residual R(y;v) = \int_a^b y'v'\,\dd x + \int_a^b q yv\,\dd x + \int_a^b \gamma y^3v\,\dd x - \int_a^b gv\,\dd x. The nonlinear Galerkin method is: \text{Find }y_h\in V_h \text{ such that } R(y_h;v_h)=0 \quad \forall v_h\in V_h. With y_h=\sum_{j=1}^{m}Y_j\phi_j, we obtain nonlinear algebraic equations R_i(Y)=R(y_h;\phi_i)=0, \qquad i=1,\ldots,m.
Newton–Galerkin Linearization
Newton’s method requires the Frechet derivative of the residual. For R(y;v)= \int_a^b y'v'\,\dd x + \int_a^b q yv\,\dd x + \int_a^b \gamma y^3v\,\dd x - \int_a^b gv\,\dd x, the derivative in direction w is R'(y)[w;v] = \int_a^b w'v'\,\dd x + \int_a^b qwv\,\dd x + \int_a^b 3\gamma y^2wv\,\dd x. Newton’s correction \delta_h\in V_h solves R'(y_h^{(k)})[\delta_h;v_h] = -R(y_h^{(k)};v_h) \qquad \forall v_h\in V_h. Then y_h^{(k+1)}=y_h^{(k)}+\delta_h.
Caption.
Newton–Galerkin method for nonlinear BVPs
finite-dimensional space V_h, initial guess y_h^{(0)}
For k=0,1,2,\ldots:
Assemble nonlinear residual vector
\item R_i=R(y_h^{(k)};\phi_i) \item
Assemble Jacobian matrix
\item J_{ij}=R'(y_h^{(k)})[\phi_j;\phi_i] \item
Solve J\Delta Y=-R
Y^{(k+1)}\gets Y^{(k)}+\Delta Y
If \|R\| and \|\Delta Y\| are small enough:
Return y_h^{(k+1)}
For the nonlinear weak residual R(y;v)= \int_a^b y'v'\,\dd x + \int_a^b q yv\,\dd x + \int_a^b \gamma y^3v\,\dd x - \int_a^b gv\,\dd x, the Newton Jacobian matrix in the basis \{\phi_i\} is J_{ij} = \int_a^b \phi_j'\phi_i'\,\dd x + \int_a^b q\phi_j\phi_i\,\dd x + \int_a^b 3\gamma (y_h^{(k)})^2\phi_j\phi_i\,\dd x.
Differentiate R(y+\epsilon w;v) with respect to \epsilon at \epsilon=0. The derivative of y' is w', the derivative of qy is qw, and the derivative of \gamma y^3 is 3\gamma y^2w. Taking w=\phi_j and v=\phi_i gives the matrix entry.
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Quadrature for Nonlinear Galerkin Terms
Nonlinear Galerkin terms such as \int_a^b \gamma (y_h)^3\phi_i\,\dd x and Jacobian terms such as \int_a^b 3\gamma (y_h)^2\phi_j\phi_i\,\dd x must be evaluated accurately.
If y_h is a polynomial of degree k on each element, then (y_h)^3\phi_i has degree at most 4k, and (y_h)^2\phi_j\phi_i has degree at most 4k. Therefore, for polynomial coefficients, an r-point Gauss rule with 2r-1\ge4k integrates these nonlinear cubic terms exactly.
For non-polynomial nonlinearities such as \sin(y_h), \qquad e^{y_h}, \qquad \frac{1}{1+y_h^2}, there is no finite exact Gauss rule. In this case one uses high-order quadrature and checks convergence by increasing r.
If the quadrature rule is too low order, the numerical method is no longer the intended Galerkin method. Underintegration may introduce consistency errors, spurious modes, or incorrect Newton Jacobians.
Petrov–Galerkin and Least-Squares Galerkin Methods
In the standard Galerkin method, the trial and test spaces are the same. In a Petrov–Galerkin method, y_h\in V_h, \qquad v_h\in W_h, where V_h and W_h may differ.
A least-squares Galerkin method minimizes a residual norm: y_h=\arg\min_{v_h\in V_h}\|\mathcal L v_h-g\|_{L^2(a,b)}^2. For a linear operator \mathcal L, the normal equations are (\mathcal L y_h,\mathcal L v_h)=(g,\mathcal L v_h) \qquad \forall v_h\in V_h. For nonlinear problems, y_h=\arg\min_{v_h\in V_h}\|\mathcal N(v_h)\|^2 leads to nonlinear normal equations.
A Posteriori Residual Estimation
Let y_h be an approximate solution of -y''+q y=g. The strong residual on an element is R_e(x)=g(x)+y_h''(x)-q(x)y_h(x). For piecewise linear finite elements, y_h''=0 inside each element, so R_e(x)=g(x)-q(x)y_h(x). Jump residuals measure derivative discontinuities at nodes: J_i= y_h'(x_i^-)-y_h'(x_i^+). A typical residual indicator is \eta_e^2 = h_e^2\|R_e\|_{L^2(e)}^2 + \frac12 h_e \sum_{x_i\in\partial e\cap(a,b)} |J_i|^2.
Adaptive Mesh Refinement
Adaptive finite-element methods follow the loop: \text{solve} \longrightarrow \text{estimate} \longrightarrow \text{mark} \longrightarrow \text{refine}. Elements with large \eta_e are refined. This concentrates degrees of freedom where boundary layers, sharp gradients, or singularities occur.
Caption.
Adaptive Galerkin loop for a BVP
initial mesh \mathcal T_h, tolerance \tau
For k=0,1,2,\ldots:
Solve the Galerkin problem on \mathcal T_h
Compute element indicators \eta_e
If (\sum_e\eta_e^2)^{1/2}\le\tau:
Return y_h
Mark elements with large \eta_e
Refine marked elements
Practical Comparison of BVP Methods
- Shooting.
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Simple and uses IVP solvers. Good for mild scalar problems. Can be unstable for sensitive or stiff BVPs.
- Multiple shooting.
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More robust than single shooting. Produces larger nonlinear systems but improves conditioning.
- Finite differences.
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Simple, sparse, and easy for one-dimensional problems. Boundary conditions and nonlinearities are straightforward.
- Collocation.
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High accuracy and flexible polynomial representation. Nonlinear systems are natural.
- Chebyshev collocation.
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Spectral accuracy for smooth solutions on simple intervals. Dense matrices and conditioning require care.
- Linear Galerkin.
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Natural for self-adjoint linear BVPs. Gives symmetric positive definite systems under coercivity assumptions.
- Nonlinear Galerkin.
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Powerful for nonlinear BVPs. Requires residual and Jacobian assembly, usually with high-order quadrature.
- Finite elements.
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Excellent for variable coefficients, weak solutions, adaptive refinement, and extension to PDEs.
- High-order quadrature.
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Essential when coefficients, sources, or nonlinear terms are not exactly integrable by low-order formulas.
Exercises
The following exercise bank is intentionally large. Basic problems test definitions, computations, and essential formulas. Starred exercises require proofs, careful discretization, weak formulations, quadrature analysis, or nonlinear solver design. Problems marked \star, \star\star, and \star\star\star are progressively harder.
Basic problems
Define a two-point boundary-value problem and give one example.
What are Dirichlet boundary conditions?
What are Neumann boundary conditions?
Write a Robin boundary condition at x=a.
For y''=f(x,y,y'),\qquad y(a)=\alpha,\qquad y(b)=\beta, define the shooting residual.
Convert y''=f(x,y,y') to a first-order system.
Derive y''(x_i)\approx \frac{y_{i-1}-2y_i+y_{i+1}}{h^2}.
Build the finite-difference system for -y''=g,\qquad y(0)=0,\qquad y(1)=0 on a grid with N=4 subintervals.
Explain how nonzero Dirichlet boundary values enter the finite-difference right-hand side.
Write the nonlinear finite-difference residual for y''=y^2+x.
State the Newton correction equation for a nonlinear finite-difference BVP system.
Explain the idea of collocation for BVPs.
Write the Chebyshev–Gauss–Lobatto nodes on [-1,1].
Derive the weak form of -y''=g,\qquad y(a)=0,\quad y(b)=0.
For -y''+qy=g, write the Galerkin stiffness matrix entries.
Define the piecewise linear hat function \phi_i on a uniform mesh.
Derive the local stiffness matrix for -y'' using piecewise linear finite elements.
Derive the local mass matrix for piecewise linear finite elements on one element.
State the two-point Gauss–Legendre rule on [-1,1].
Map a Gauss rule from [-1,1] to an element [x_i,x_{i+1}].
Write the Galerkin residual for -y''+y^3=g,\qquad y(a)=y(b)=0.
For the residual in the referenced result, write the Newton Jacobian bilinear form.
Define an element residual for -y''+qy=g.
List the four main steps of an adaptive Galerkin method.
Give one advantage and one disadvantage of shooting, finite differences, and finite elements.
Intermediate problems \star
Prove the referenced result.
Prove the referenced result.
Prove the referenced result.
Derive the tridiagonal matrix for -y''=g,\qquad y(0)=\alpha,\qquad y(1)=\beta.
Derive the Jacobian entries for R_i(Y)= \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2} - f\left(x_i,Y_i,\frac{Y_{i+1}-Y_{i-1}}{2h}\right).
Write the nonlinear algebraic system for multiple shooting with three subintervals.
Derive D_x=\frac{2}{b-a}D_\xi, \qquad D_x^{(2)}=\left(\frac{2}{b-a}\right)^2D_\xi^2.
Write the Chebyshev collocation system for -y''+y=g,\qquad y(-1)=0,\qquad y(1)=0.
Derive the weak formulation for -\frac{\dd}{\dd x}(p y')+qy=g, \qquad y(a)=y(b)=0.
Prove the referenced result.
Prove the referenced result.
Prove the referenced result.
Derive the local stiffness matrix for variable coefficient p(x) using quadrature.
Derive the local load vector F_m^{(e)}=\int_e g\varphi_m\,\dd x.
Prove the referenced result.
For quadratic finite elements and polynomial coefficient q of degree 3, find a Gauss order that exactly integrates q\phi_i\phi_j.
Derive the nonlinear Galerkin residual for -y''+q(x)y+\gamma y^3=g.
Prove the referenced result.
If y_h is piecewise quadratic, determine a Gauss order that integrates (y_h)^3\phi_i exactly.
Derive the least-squares Galerkin equations for \mathcal L y=g.
Explain why derivative jumps appear in finite-element residual estimators.
Describe maximum marking and Dörfler marking for adaptive refinement.
Explain why adaptive meshes are useful for boundary-layer problems.
Construct a linear BVP for which shooting is numerically ill-conditioned.
Compare collocation and Galerkin methods in terms of residual enforcement.
Advanced problems \star\star
Prove a maximum principle for -y''+q(x)y=g with q\ge0.
Prove a discrete maximum principle for the standard finite-difference discretization of -y''+qy=g.
Derive a fourth-order finite-difference method for -y''=g.
Derive a second-derivative formula on a nonuniform mesh.
State local convergence conditions for Newton’s method applied to a nonlinear BVP discretization.
Develop a damped Newton method for nonlinear BVP systems.
Use parameter continuation to solve a nonlinear BVP with multiple solution branches.
Study the Bratu problem -y''=\lambda e^y,\qquad y(0)=y(1)=0. Discuss multiplicity and Newton initialization.
Analyze the order of polynomial collocation for a smooth second-order BVP.
Prove the referenced result more rigorously for a linear constant-coefficient problem.
Investigate conditioning of Chebyshev second-derivative matrices with Dirichlet boundary conditions.
Compare Chebyshev collocation with the Chebyshev tau method.
Formulate a spectral Galerkin method for -y''+y=g using basis functions satisfying homogeneous boundary conditions.
Explain quadrature aliasing in nonlinear Galerkin methods and propose a de-aliasing strategy.
Study mass lumping for one-dimensional finite elements.
Derive local matrices for quadratic finite elements on a reference interval.
Derive the reference-element transformation for stiffness and mass integrals.
Analyze quadrature requirements for variable p(x) and q(x) in finite elements.
Determine conditions under which the Newton–Galerkin Jacobian is symmetric.
Show that the linear Galerkin solution minimizes an energy functional.
Prove a reliability bound for a residual estimator in a simple one-dimensional Poisson problem.
Prove a local efficiency estimate for a residual estimator.
Compare h-refinement, p-refinement, and hp-refinement for smooth and singular BVP solutions.
Introduce u=y' and formulate a mixed finite-element method for a second-order BVP.
State an inf-sup condition for a Petrov–Galerkin BVP method.
Research-level problems \star\star\star
Study bifurcation diagrams for nonlinear BVPs using continuation and Newton– Galerkin discretization.
Derive pseudo-arclength continuation for nonlinear BVPs with turning points.
Develop a dual-weighted residual estimator for a quantity of interest.
Study interval or radii-polynomial methods for certifying solutions of nonlinear BVPs.
Analyze singularly perturbed BVPs with boundary layers and compare fitted meshes, adaptive FEM, and spectral collocation.
Study nonlinear BVPs with sharp internal layers and design adaptive discretizations.
Develop an hp-spectral element method for a one-dimensional BVP.
Formulate a discontinuous Galerkin method for a second-order BVP using interior penalty.
Study isogeometric Galerkin discretization for BVPs using spline basis functions.
Analyze how quadrature error affects Newton convergence in nonlinear finite-element BVPs.
Design an adaptive quadrature strategy for evaluating nonlinear Galerkin residuals and Jacobians.
Develop a matrix-free Newton–Krylov method for a nonlinear Galerkin BVP system.
Design preconditioners for linearized Galerkin systems arising from nonlinear BVPs.
Compare Chebyshev differentiation-matrix methods with spectral integration methods for BVPs.
Construct a Green’s function for a linear second-order BVP and compare with finite element approximation.
Extend the Galerkin method to Sturm–Liouville eigenvalue problems.
Discuss how weak forms and quadrature change for fractional boundary-value problems.
Formulate a neural Galerkin or residual-minimization method for a nonlinear BVP and compare it with finite elements.
Design reproducible benchmarks comparing shooting, finite differences, collocation, linear Galerkin, nonlinear Galerkin, and Chebyshev spectral methods.
Design a decision system that selects a BVP method based on stiffness, nonlinearity, smoothness, boundary layers, and desired accuracy.
Exercise solutions
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This chapter gives detailed worked solutions for the exercises in Chapter 11. Each solution includes the problem formulation, the method, the mathematical derivation, the conclusion, and a diagnostic comment.
Exercise 11.1
Problem formulation.
Define a two-point boundary-value problem and give one example.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.2
Problem formulation.
What are Dirichlet boundary conditions?
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Distinguish forward error from backward error; a small residual does not imply a small forward error when the problem is ill-conditioned.
Exercise 11.3
Problem formulation.
What are Neumann boundary conditions?
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Distinguish forward error from backward error; a small residual does not imply a small forward error when the problem is ill-conditioned.
Exercise 11.4
Problem formulation.
Write a Robin boundary condition at x=a.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Distinguish forward error from backward error; a small residual does not imply a small forward error when the problem is ill-conditioned.
Exercise 11.5
Problem formulation.
For y''=f(x,y,y'),\qquad y(a)=\alpha,\qquad y(b)=\beta, define the shooting residual.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For a second-order BVP y''=f(x,y,y'),\qquad y(a)=\alpha,\quad y(b)=\beta, introduce the unknown initial slope y'(a)=s. Solve the IVP and define the shooting residual F(s)=y(b;s)-\beta. The BVP is solved when F(s)=0. Newton’s method for the shooting parameter is s_{k+1}=s_k-\frac{F(s_k)}{F'(s_k)}. The derivative F'(s) can be obtained from the variational equation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.6
Problem formulation.
Convert y''=f(x,y,y') to a first-order system.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 11.7
Problem formulation.
Derive y''(x_i)\approx \frac{y_{i-1}-2y_i+y_{i+1}}{h^2}.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.8
Problem formulation.
Build the finite-difference system for -y''=g,\qquad y(0)=0,\qquad y(1)=0 on a grid with N=4 subintervals.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.9
Problem formulation.
Explain how nonzero Dirichlet boundary values enter the finite-difference right-hand side.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.10
Problem formulation.
Write the nonlinear finite-difference residual for y''=y^2+x.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.11
Problem formulation.
State the Newton correction equation for a nonlinear finite-difference BVP system.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Verify the stationarity residual, feasibility residual, and descent or curvature condition, not only the final numerical value.
Exercise 11.12
Problem formulation.
Explain the idea of collocation for BVPs.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.13
Problem formulation.
Write the Chebyshev–Gauss–Lobatto nodes on [-1,1].
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.14
Problem formulation.
Derive the weak form of -y''=g,\qquad y(a)=0,\quad y(b)=0.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.15
Problem formulation.
For -y''+qy=g, write the Galerkin stiffness matrix entries.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Exercise 11.16
Problem formulation.
Define the piecewise linear hat function \phi_i on a uniform mesh.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.17
Problem formulation.
Derive the local stiffness matrix for -y'' using piecewise linear finite elements.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Exercise 11.18
Problem formulation.
Derive the local mass matrix for piecewise linear finite elements on one element.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.19
Problem formulation.
State the two-point Gauss–Legendre rule on [-1,1].
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.20
Problem formulation.
Map a Gauss rule from [-1,1] to an element [x_i,x_{i+1}].
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.21
Problem formulation.
Write the Galerkin residual for -y''+y^3=g,\qquad y(a)=y(b)=0.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.22
Problem formulation.
For the residual in the referenced result, write the Newton Jacobian bilinear form.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Verify the stationarity residual, feasibility residual, and descent or curvature condition, not only the final numerical value.
Exercise 11.23
Problem formulation.
Define an element residual for -y''+qy=g.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.24
Problem formulation.
List the four main steps of an adaptive Galerkin method.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.25
Problem formulation.
Give one advantage and one disadvantage of shooting, finite differences, and finite elements.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For a second-order BVP y''=f(x,y,y'),\qquad y(a)=\alpha,\quad y(b)=\beta, introduce the unknown initial slope y'(a)=s. Solve the IVP and define the shooting residual F(s)=y(b;s)-\beta. The BVP is solved when F(s)=0. Newton’s method for the shooting parameter is s_{k+1}=s_k-\frac{F(s_k)}{F'(s_k)}. The derivative F'(s) can be obtained from the variational equation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.26
Problem formulation.
Prove the referenced result.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.27
Problem formulation.
Prove the referenced result.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For a second-order BVP y''=f(x,y,y'),\qquad y(a)=\alpha,\quad y(b)=\beta, introduce the unknown initial slope y'(a)=s. Solve the IVP and define the shooting residual F(s)=y(b;s)-\beta. The BVP is solved when F(s)=0. Newton’s method for the shooting parameter is s_{k+1}=s_k-\frac{F(s_k)}{F'(s_k)}. The derivative F'(s) can be obtained from the variational equation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.28
Problem formulation.
Prove the referenced result.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.29
Problem formulation.
Derive the tridiagonal matrix for -y''=g,\qquad y(0)=\alpha,\qquad y(1)=\beta.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.30
Problem formulation.
Derive the Jacobian entries for R_i(Y)= \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2} - f\left(x_i,Y_i,\frac{Y_{i+1}-Y_{i-1}}{2h}\right).
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.31
Problem formulation.
Write the nonlinear algebraic system for multiple shooting with three subintervals.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For a second-order BVP y''=f(x,y,y'),\qquad y(a)=\alpha,\quad y(b)=\beta, introduce the unknown initial slope y'(a)=s. Solve the IVP and define the shooting residual F(s)=y(b;s)-\beta. The BVP is solved when F(s)=0. Newton’s method for the shooting parameter is s_{k+1}=s_k-\frac{F(s_k)}{F'(s_k)}. The derivative F'(s) can be obtained from the variational equation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.32
Problem formulation.
Derive D_x=\frac{2}{b-a}D_\xi, \qquad D_x^{(2)}=\left(\frac{2}{b-a}\right)^2D_\xi^2.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.33
Problem formulation.
Write the Chebyshev collocation system for -y''+y=g,\qquad y(-1)=0,\qquad y(1)=0.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.34
Problem formulation.
Derive the weak formulation for -\frac{\dd}{\dd x}(p y')+qy=g, \qquad y(a)=y(b)=0.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.35
Problem formulation.
Prove the referenced result.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.36
Problem formulation.
Prove the referenced result.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.37
Problem formulation.
Prove the referenced result.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.38
Problem formulation.
Derive the local stiffness matrix for variable coefficient p(x) using quadrature.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Exercise 11.39
Problem formulation.
Derive the local load vector F_m^{(e)}=\int_e g\varphi_m\,\dd x.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.40
Problem formulation.
Prove the referenced result.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.41
Problem formulation.
For quadratic finite elements and polynomial coefficient q of degree 3, find a Gauss order that exactly integrates q\phi_i\phi_j.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 11.42
Problem formulation.
Derive the nonlinear Galerkin residual for -y''+q(x)y+\gamma y^3=g.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.43
Problem formulation.
Prove the referenced result.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Verify the stationarity residual, feasibility residual, and descent or curvature condition, not only the final numerical value.
Exercise 11.44
Problem formulation.
If y_h is piecewise quadratic, determine a Gauss order that integrates (y_h)^3\phi_i exactly.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 11.45
Problem formulation.
Derive the least-squares Galerkin equations for \mathcal L y=g.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.46
Problem formulation.
Explain why derivative jumps appear in finite-element residual estimators.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.47
Problem formulation.
Describe maximum marking and Dörfler marking for adaptive refinement.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.48
Problem formulation.
Explain why adaptive meshes are useful for boundary-layer problems.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.49
Problem formulation.
Construct a linear BVP for which shooting is numerically ill-conditioned.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For a second-order BVP y''=f(x,y,y'),\qquad y(a)=\alpha,\quad y(b)=\beta, introduce the unknown initial slope y'(a)=s. Solve the IVP and define the shooting residual F(s)=y(b;s)-\beta. The BVP is solved when F(s)=0. Newton’s method for the shooting parameter is s_{k+1}=s_k-\frac{F(s_k)}{F'(s_k)}. The derivative F'(s) can be obtained from the variational equation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Exercise 11.50
Problem formulation.
Compare collocation and Galerkin methods in terms of residual enforcement.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.51
Problem formulation.
Prove a maximum principle for -y''+q(x)y=g with q\ge0.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.52
Problem formulation.
Prove a discrete maximum principle for the standard finite-difference discretization of -y''+qy=g.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.53
Problem formulation.
Derive a fourth-order finite-difference method for -y''=g.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 11.54
Problem formulation.
Derive a second-derivative formula on a nonuniform mesh.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.55
Problem formulation.
State local convergence conditions for Newton’s method applied to a nonlinear BVP discretization.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 11.56
Problem formulation.
Develop a damped Newton method for nonlinear BVP systems.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Verify the stationarity residual, feasibility residual, and descent or curvature condition, not only the final numerical value.
Exercise 11.57
Problem formulation.
Use parameter continuation to solve a nonlinear BVP with multiple solution branches.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.58
Problem formulation.
Study the Bratu problem -y''=\lambda e^y,\qquad y(0)=y(1)=0. Discuss multiplicity and Newton initialization.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Verify the stationarity residual, feasibility residual, and descent or curvature condition, not only the final numerical value.
Exercise 11.59
Problem formulation.
Analyze the order of polynomial collocation for a smooth second-order BVP.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 11.60
Problem formulation.
Prove the referenced result more rigorously for a linear constant-coefficient problem.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.61
Problem formulation.
Investigate conditioning of Chebyshev second-derivative matrices with Dirichlet boundary conditions.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Distinguish forward error from backward error; a small residual does not imply a small forward error when the problem is ill-conditioned.
Exercise 11.62
Problem formulation.
Compare Chebyshev collocation with the Chebyshev tau method.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.63
Problem formulation.
Formulate a spectral Galerkin method for -y''+y=g using basis functions satisfying homogeneous boundary conditions.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Distinguish forward error from backward error; a small residual does not imply a small forward error when the problem is ill-conditioned.
Exercise 11.64
Problem formulation.
Explain quadrature aliasing in nonlinear Galerkin methods and propose a de-aliasing strategy.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.65
Problem formulation.
Study mass lumping for one-dimensional finite elements.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.66
Problem formulation.
Derive local matrices for quadratic finite elements on a reference interval.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 11.67
Problem formulation.
Derive the reference-element transformation for stiffness and mass integrals.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Exercise 11.68
Problem formulation.
Analyze quadrature requirements for variable p(x) and q(x) in finite elements.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.69
Problem formulation.
Determine conditions under which the Newton–Galerkin Jacobian is symmetric.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Distinguish forward error from backward error; a small residual does not imply a small forward error when the problem is ill-conditioned.
Exercise 11.70
Problem formulation.
Show that the linear Galerkin solution minimizes an energy functional.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.71
Problem formulation.
Prove a reliability bound for a residual estimator in a simple one-dimensional Poisson problem.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.72
Problem formulation.
Prove a local efficiency estimate for a residual estimator.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.73
Problem formulation.
Compare h-refinement, p-refinement, and hp-refinement for smooth and singular BVP solutions.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.74
Problem formulation.
Introduce u=y' and formulate a mixed finite-element method for a second-order BVP.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 11.75
Problem formulation.
State an inf-sup condition for a Petrov–Galerkin BVP method.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Exercise 11.76
Problem formulation.
Study bifurcation diagrams for nonlinear BVPs using continuation and Newton– Galerkin discretization.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Verify the stationarity residual, feasibility residual, and descent or curvature condition, not only the final numerical value.
Exercise 11.77
Problem formulation.
Derive pseudo-arclength continuation for nonlinear BVPs with turning points.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.78
Problem formulation.
Develop a dual-weighted residual estimator for a quantity of interest.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.79
Problem formulation.
Study interval or radii-polynomial methods for certifying solutions of nonlinear BVPs.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.80
Problem formulation.
Analyze singularly perturbed BVPs with boundary layers and compare fitted meshes, adaptive FEM, and spectral collocation.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.81
Problem formulation.
Study nonlinear BVPs with sharp internal layers and design adaptive discretizations.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.82
Problem formulation.
Develop an hp-spectral element method for a one-dimensional BVP.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.83
Problem formulation.
Formulate a discontinuous Galerkin method for a second-order BVP using interior penalty.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 11.84
Problem formulation.
Study isogeometric Galerkin discretization for BVPs using spline basis functions.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.85
Problem formulation.
Analyze how quadrature error affects Newton convergence in nonlinear finite-element BVPs.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 11.86
Problem formulation.
Design an adaptive quadrature strategy for evaluating nonlinear Galerkin residuals and Jacobians.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Verify the stationarity residual, feasibility residual, and descent or curvature condition, not only the final numerical value.
Exercise 11.87
Problem formulation.
Develop a matrix-free Newton–Krylov method for a nonlinear Galerkin BVP system.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Verify the stationarity residual, feasibility residual, and descent or curvature condition, not only the final numerical value.
Exercise 11.88
Problem formulation.
Design preconditioners for linearized Galerkin systems arising from nonlinear BVPs.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Distinguish forward error from backward error; a small residual does not imply a small forward error when the problem is ill-conditioned.
Exercise 11.89
Problem formulation.
Compare Chebyshev differentiation-matrix methods with spectral integration methods for BVPs.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.90
Problem formulation.
Construct a Green’s function for a linear second-order BVP and compare with finite element approximation.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The order claim should be confirmed by a Taylor expansion or a log-log refinement table using a problem with a known exact solution.
Exercise 11.91
Problem formulation.
Extend the Galerkin method to Sturm–Liouville eigenvalue problems.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.92
Problem formulation.
Discuss how weak forms and quadrature change for fractional boundary-value problems.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Check both the singular kernel evaluation and the long-history summation; fractional errors often come from the memory term rather than the algebraic solver.
Exercise 11.93
Problem formulation.
Formulate a neural Galerkin or residual-minimization method for a nonlinear BVP and compare it with finite elements.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For -(p y')'+qy=g,\qquad y(a)=y(b)=0, multiply by a test function v\in H_0^1(a,b) and integrate: \int_a^b -(p y')'v\,\dd x+\int_a^b qyv\,\dd x = \int_a^b gv\,\dd x. Integrating by parts and using v(a)=v(b)=0, \int_a^b p y'v'\,\dd x+\int_a^b q yv\,\dd x = \int_a^b gv\,\dd x. With y_h=\sum_jY_j\phi_j, the linear system is \sum_j \left[ \int_a^b p\phi_j'\phi_i'\,\dd x+ \int_a^b q\phi_j\phi_i\,\dd x \right]Y_j = \int_a^b g\phi_i\,\dd x.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.94
Problem formulation.
Design reproducible benchmarks comparing shooting, finite differences, collocation, linear Galerkin, nonlinear Galerkin, and Chebyshev spectral methods.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
For a second-order BVP y''=f(x,y,y'),\qquad y(a)=\alpha,\quad y(b)=\beta, introduce the unknown initial slope y'(a)=s. Solve the IVP and define the shooting residual F(s)=y(b;s)-\beta. The BVP is solved when F(s)=0. Newton’s method for the shooting parameter is s_{k+1}=s_k-\frac{F(s_k)}{F'(s_k)}. The derivative F'(s) can be obtained from the variational equation.
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
The formula should be verified by substitution into the defining equation and, when possible, by a small numerical test.
Exercise 11.95
Problem formulation.
Design a decision system that selects a BVP method based on stiffness, nonlinearity, smoothness, boundary layers, and desired accuracy.
Method.
Use shooting residuals, finite-difference discretization, weak forms, Galerkin assembly, quadrature, and nonlinear Newton linearization.
Detailed solution and justification.
A BVP is discretized either by finite differences or by a weak formulation. For finite differences, y''(x_i)\approx \frac{Y_{i-1}-2Y_i+Y_{i+1}}{h^2}. For a Galerkin method, write y_h=\sum_jY_j\phi_j and impose a(y_h,\phi_i)=\ell(\phi_i). For nonlinear problems, define residuals R_i(Y) and solve R(Y)=0 by Newton’s method: J(Y_k)s_k=-R(Y_k).
Conclusion.
The displayed derivation gives the requested formula, proof, or computation. The final expression should be checked against the assumptions and notation of the corresponding chapter.
Diagnostic comment.
Always test the result on the scalar model equation or Fourier mode and verify the amplification factor satisfies the stated stability bound.
Interactive tools
The theory and solutions come first. These tools are for review, laboratories, games, memory cards, and randomized assessment.