# Which 2D PDE with an exact solution can I use to test/verify my FEM-PDE code?

I have created a program to solve 2D, time-dependent PDEs with the finite element method and get reasonable looking results for the 2D acoustic wave equation. Now I would like to go further and solve a PDE with a known exact/analytic solution to compare against. However, I have a lot of trouble finding a suitable equation. It seems that there are no 2D, time-dependent equations with an analytic solution that does not involve infinite sums or the like. I thought about taking a 1D equation and extend it to two dimensions by just solving it on a 2D domain without changing anything, in effect replicating the equation along the y-axis. However, boundary conditions seem to mess things up and these 1D equations work on infinite domains, i.e. the whole real axis.

Is there such an equation for me to use? Is there a different established way of testing the correctness of a PDE solver implementation?

• For a simple domain (square, circle) there are tons of analytic solutions. You can produce one easily: just make up some function f(x,y,t) and plug it in your PDE, so you can find what source term on the RHS is needed to make this f(x,y,t) satisfy the equation. That's the idea, you can look up the details and examples if you search for "manufactured solutions". Nov 23, 2020 at 23:53