Math 6610 • Analysis of Numerical Methods I

{Elliptic Equations} Consider the standard five-point difference
approximation (centered difference for both the gradient and divergence operators)
for the variable coefficient Poisson's equation
\[
  -\nabla\cdot(a\nabla u)=f
\]
with Dirichlet boundary conditions, in a two-dimensional rectangular region. We assume
that \(a(\mathbf{x})\ge \delta>0\). The approximate solution \(\{u_{i,j}\}\) satisfies a
linear system \(Au=b\).

\textbf{(a)} State and prove the maximum principle for the numerical solution
\(u_{i,j}\).

\textbf{(b)} Use the maximum principle to show that the coefficient matrix \(A\) is
nonsingular.

\textbf{(c)} Derive the matrix \(A\) in the one-dimensional case and show that it is
symmetric and positive definite.

\textbf{(d)} With this linear system, do we get convergence with Jacobi iterations?
What about Gauss-Seidel iterations? Give reasons for your answers.