Chapter 15.1, Problem 7E

1. (a) In Problem 12 of Exercises 12.6 you solved a potential problem using a special form of Poisson’s equation $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=f(x,y)$. Show that the difference equation replacement for Poisson’s equation is

$u_{i+1}{}_{.j}+u_{i,}{}_{j+1}+u_{i+1}{}_{,j}+u_i{}_{,j-1}-4u_{ij}=h^{2}f(x,y).$

1. (b) Use the result in part (a) to approximate the solution of the Poisson equation $\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=-2$ at the interior points of the region in Figure 15.1.7. The mesh size is $h=\frac{1}{2},u=1$ at every point along ABCD, and u = 0 at every point along DEFGA. Use symmetry and, if necessary, Gauss-Seidel iteration.

FIGURE 15.1.7 Región for Problem 7