Math 6610 • Analysis of Numerical Methods I

{Elliptic Problems:}

Consider the standard five-point difference approximation (centered difference
for both the gradient and divergence operators) for the variable coefficient
Poisson equation
\[
  -\nabla\cdot(a\nabla v)=f
\]
with Dirichlet boundary conditions, in a two-dimensional rectangular region. We
assume that \(a(x,y)\geq a_0>0\). The approximate solution \(\{u_{i,j}\}\) satisfies
a linear system \(Au=b\).

\textbf{1.} State and prove the maximum principle for the numerical solution
\(u_{i,j}\).

\textbf{2.} Derive the matrix \(A\) in the one-dimensional case and show that it
is symmetric and positive definite.

\textbf{3.} For the one-dimensional and constant-coefficient case, show that
the global error \(e_j=v(x_j)-u_j\) satisfies \(\|e\|_2=O(h^2)\) as the space step
\(h\to0\).

\textbf{4.} Discuss the advantages and disadvantages of trying to solve the
system for the two-dimensional problem using \textbf{(i)} the SOR (Successive
Over Relaxation) method and \textbf{(ii)} the (preconditioned) Conjugate
Gradient method.

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