Math 6610 • Analysis of Numerical Methods I

{Elliptic Problems:}

For the one dimensional Poisson problem for \(v(x)\)
\[
  -v''(x)=f(x)
\]
with Dirichlet boundary conditions in the interval \([0,1]\), consider the scheme
\[
  -(D_+D_-U)_j\equiv -U_{j-1}+2\,U_j-U_{j+1}=F_j
\]
for \(j=1,2,\ldots,N-1\) where \(Nh=1\), \(F_j\equiv h^2f(jh)\), and
\(U_0=U_N=0\). The approximate solution satisfies a linear system \(AU=b\), where
\(U=(U_1,U_2,\ldots,U_{N-1})^T\) and \(b=(F_1,F_2,\ldots,F_{N-1})^T\).

\partitem{a} State and prove the maximum principle for the numerical solution
\(U_j\).

\partitem{b} Derive the matrix \(A\) and show that it is symmetric and positive
definite.

\partitem{c} Show that the global error \(e_j=v(x_j)-U_j\) satisfies
\(\|e\|_\infty=O(h^2)\) as the space step \(h\to0\).

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