Past exam of the numerical solution of differential equations course of the University of Oxford / 2010 / B / 6

With zero boundary values eliminated, the diagonal of $L_h$ is $-4I/h^2$. The Jacobi method is thereforewhich averages the four neighboring components. The sine addition identity shows that the supplied product vectors are eigenvectors, withThe discrete sine vectors form a basis, so these are all eigenvalues. Their largest modulus isattained at the lowest and highest sine frequencies. Thus the asymptotic error reduction factor per iteration is $\cos (\pi h)$. If the convergence rate is expressed logarithmically, $R=-\log \rho (J)\sim {\pi }^2{h}^2/2$ as $h\to 0$. A fixed error reduction therefore takes $O(h^{-2})$ Jacobi sweeps. For the special grid $m=2$, $J=0$ and one sweep solves its single interior equation.

The endpoint $\alpha =0$ in the printed nonnegative-parameter hypothesis must be excluded. We first establish the intended result for $\alpha >0$, then describe that endpoint exactly.

Write $X,Y$ for the two directional matrices; they commute because they act on separate tensor-product coordinates. At a limit $V$, the intermediate vector also has a limit $\tilde{V}$. Subtract the two limiting equations in the form that cancels $X\tilde{V}$ and $YV$; this gives $2\alpha (\tilde{V}-V)=0$. Hence $\tilde{V}=V$ for $\alpha >0$. Substitution then gives $(X+Y)V=h^{2}f$, precisely the required discrete equation.

Eliminating the intermediate vector gives the alternating direction implicit iteration matrixThe sine-product basis simultaneously diagonalizes $X,Y$, with eigenvalues $-{\xi }_p$ and $-{\eta }_q$, whereThusFor $\alpha >0$, each factor has modulus strictly less than one. Hence $\rho (G)<1$, proving convergence from every starting vector to the unique discrete solution.

At $\alpha =0$ the two factors are both $-1$, so $G=I$. The forcing terms cancel in the composed iteration as well, and $V^{k+1}=V^k$ for every starting vector. An incorrect initial vector therefore remains incorrect forever, even though the sequence is constant and has a limit. This disproves both the literal fixed-point implication and the claimed general convergence-to-solution assertion at zero; strict positivity of $\alpha$ is the necessary repair.