Problem

Solve Poisson's equation on the unit square (square.m) with homogeneous Dirichlet boundary conditions, $a=1$, and the right-hand side given by

$$f(x) = 5 \pi^2 \sin(\pi x_1) \sin(2\pi x_2).$$ (3)

Compare your computed solution $U(x)$ with the exact solution $u(x)$. How large is the error in the maximum norm? The maximum norm of the error is given by

$$\Vert e\Vert _{\infty} = \Vert U - u\Vert _{\infty} = \max_{x\in\Omega} \vert U(x) - u(x)\vert.$$ (4)

How large is the error when you refine the mesh (square_refined.m)?

Check your answer: The error in the maximum norm should be $0.0216$ for the first mesh and $0.0055$ for the refined mesh. Notice that the error is decreased by a factor $4 = 2^2$ when we decrease the mesh size with a factor $2$.