Consider following elliptic boundary value problem:

$$u $=2\backslash$pi $2$ sin$(\backslash$pi x$)$ cos$(\backslash$pi y$),($x$,$ y$)$ in $(0,1)2,$

subject to the boundary condition

u$($x$,0)=$ sin$(\backslash$pi x$),$ u$($x$,1)=$ sin$(\backslash$pi x$),$ x in $[0,1],$

u$(0,$y$)=$ u$(1,$y$)=0,$ y in $[0,1].$

Implement a second order finite difference scheme for this problem on the $($N $+1)\backslash$times $($N $+1)$ uniform grid $($same as described in $[$T$4.]).$ Compare your numerical solution with the exact solution u$($x$,$ y$)=$ sin$(\backslash$pi x$)$ cos$(\backslash$pi y$).$ Try a few different meshes N $=20,40,80,160$ and compute the corresponding errors in discrete L$1,$ L$\backslash$infty $,$ L$2$ and H$1$ norms. Plot all these errors in the same figure vs h $=1/($N $+1)$ using loglog scale and verify the second order accuracy. $($You may use a direct method or iterative method to solve the resulting linear system.$)$