Chapter 8.3, Problem 14CP

Solve the elliptic partial differential equations with Dirichlet boundary conditions by the Finite Element Method with $h=k=0.1$ . Plot the solution.

$\begin{array}{l}\text{a}\text{. }\{\begin{array}{c}\Delta u+\sin \pi xy=\left(x^2+y^2\right)u\\ u(x,1)=0\text{ for 0}\le x\le 1\\ u(x,1)=0\text{ for 0}\le x\le 1\\ \begin{array}{l}u(0,y)=0\text{ for 0}\le y\le 1\\ u(1,y)=0\text{ for 0}\le y\le 1\end{array}\end{array}\\ \text{b}\text{. }\{\begin{array}{c}\Delta u+(\sin \pi xy)u=e^{2xy}\\ u(x,0)=0\text{ for 1}\le x\le 1\\ u(x,1)=0\text{ for 0}\le x\le 1\\ \begin{array}{l}u(0,y)=0\text{ for 0}\le y\le 1\\ u(1,y)=0\text{ for 0}\le y\le 1\end{array}\end{array}\end{array}$