I have a problem that solving the Poisson equation with kink ( discontinuous gradient but solution is continuous ) in the analytical solution, I want to solve this problem with FEM. To approximate this problem with continuous Galerkin, very fine resolution is required to capture the sharp gradient jump. With the attempt to solve it with less computation, I go through the discussion in the community and found some very useful inofrmation like
I alsoe tried out solving it with discontinuous Galerkin or add some penalty stabilization term on the standard Galerkin form, but still cannot get an idea approximation.
So my question is, what is the best method that I can rely on to solve this specific problem, that in this Poisson equation, the kink of the solution always aligned with the mesh facet.
Thanks a lot for your comments and help!