How to solve the Poisson equation with KINK aligned with mesh facet

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

Why is my second order accurate method only converging at first order when the coefficients are rough?

and

Example of a continuous function that is difficult to approximate with polynomials

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!

• I don't think I quite understand your question since you already state the answer: You need to align the mesh to the kink. Or are you asking how one can choose a mesh that aligns with the kink? – Wolfgang Bangerth Jun 21 at 18:41