I have a question about discretization of 2-D Poisson. For an equation such as $\Delta u=sin(x,y)$, all boundary conditions are given like $u ( x , 0 ) = u ( x , \pi ) =$$u ( 0 , y ) = u ( \pi , y ) = 0$. If we discretize it in the domain, is there a way to obtain exact solution at grid points(like exact solution at u(1,1))?
Any help would be helpful. Thanks in advance.