Laplace's and Poisson's Equations
Laplace's equation reads
$$
\begin{equation*}
\nabla^2 u(\mathbf{x})=u_{xx}+u_{yy}=0.
\end{equation*}
$$
with possible boundary conditions
\( u(x,y) = g(x,y) \) on the border \( \delta\Omega \). There is no time-dependence.
We seek a solution in the region \( \Omega \) and we choose a quadratic mesh
with equally many steps in both directions. We could choose the grid to be rectangular or following
polar coordinates \( r,\theta \) as well. Here we choose equal steps lengths in the \( x \) and
the \( y \) directions. We set
$$
\begin{equation*} h=\Delta x = \Delta y = \frac{L}{n+1},\end{equation*}
$$
where \( L \) is the length of the sides and we have \( n+1 \) points in both directions.