Chapter 15.1, Problem 8E

Use the result in part (a) of Problem 7 to approximate the solution of the Poisson equation

$\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=-64$

at the interior points of the region in Figure 15.1.8. The mesh size is $h=\frac{1}{8}$, and u = 0 at every point on the boundary of the region. If necessary, use Gauss-Seidel iteration.

FIGURE 15.1.8 Region for Problem 8

7. (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

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