Math 6610 • Analysis of Numerical Methods I

{Elliptic Problems:}

For the one dimensional Poisson problem for \(v(x)\)
\[
  -v''(x)+\alpha v(x)=f(x),
\]
where \(\alpha\ge0\) is constant, along with Dirichlet boundary conditions in the
interval \([0,1]\), consider the scheme
\[
  \Delta_h U_j\equiv
  \frac{1}{h^2}\left(-U_{j-1}+2U_j-U_{j+1}\right)=f_j
\]
for \(j=1,2,\ldots,N-1\) where \(Nh=1\), \(f_j\equiv f(jh)\), 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
\(b=h^2(f_1,f_2,\ldots,f_{N-1})^T\).

\partitem{a} State and prove the maximum principle for any grid function
\(V=\{V_j\}\) with values for \(j=0,1,\ldots,N\), that satisfies
\(\Delta_h V_j\ge0\).

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

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