Math 6620 • Analysis of Numerical Methods II

Consider the 9-point discrete Laplacian
\[
(\Delta^h_9 U)_{j,l} =
\frac{1}{6h^2}\left(
U_{j-1,l-1}+U_{j+1,l-1}+U_{j-1,l+1}+U_{j+1,l+1}
+4U_{j,l-1}+4U_{j-1,l}+4U_{j+1,l}+4U_{j,l+1}-20U_{j,l}
\right)
\]

\begin{enumerate}

\item
State and prove a maximum principle for \(\Delta^h_9 U\). (Hint: Think about what the key observation
is in stating and proving a maximum principle for the standard 5-point discrete Laplacian.)


\item
Show that the following problem has a solution and that it is unique:
\begin{equation}
\tag{2}
\begin{aligned}
\frac{1}{6h^2}\big(&U_{j-1,l-1}+U_{j+1,l-1}+U_{j-1,l+1}+U_{j+1,l+1}\\
&+4U_{j,l-1}+4U_{j-1,l}+4U_{j+1,l}+4U_{j,l+1}-20U_{j,l}\big)
= f_{j,l},
\end{aligned}
\end{equation}
for \(j = 1,\ldots,m\) and \(l = 1,\ldots,m\) (where \((m+1)h = 1\)) along with the conditions \(U_{j,l} = 0\) at grid
points on the boundary of the domain.


\item
The local trucation error in using the scheme in Eq. (2) to find an approximate solution of
\(\Delta u = f\) for \(0 < x < 1\) and \(0 < y < 1\) with homogeneous Dirichlet boundary conditions is given by
\[
\tau_{j,l} = \frac{1}{12}(u_{xxxx}+u_{xxyy}+u_{yyyy})h^2 + O(h^4),
\]
where the derivatives of \(u\) are evaluated at \(x_j = jh\), \(y_l = lh\). Prove that the values \(U_{j,l}\) obtained
from Equations (2), along with zero discrete boundary values, converge to the values \(u(x_j,y_l)\) of
the true solution to the PDE BVP as \(h \to 0\). (Hint: Use the relation between the set of global
error values \(E_{j,l} = U_{j,l} - u(x_j,y_l)\) and the values of the local truncation error.)

\end{enumerate}