I was trying to obtain analytical solution of 2D Laplace equation. I understood the standard procedure which involved the Method of separation of variable.
I was trying to obtain the exact solution of Laplace equation with these BCs.
u(0,y)=0, u(1,y)=0, u(x,0)=1, u(x,1)=1. The domain is a unit square and FEM solution looks similar to this one (x limit is different in the figure).
The solution contains discontinuity at the corners. This indicates that the exact solution should be a piecewise function. How do I even approach this problem to obtain an exact piecewise solution?
I found several publications where people have used exact solution for 2D diffusion equation with discontinuous coefficients.