I'm working with an elliptic reaction diffusion PDE of the form
I've noticed that when the reaction term dominates over the diffusion (i.e. $c>>k$), the true (exact) solution has sharp gradients near the boundaries. I've also noticed that my numerical approximations via finite difference, finite volume, and standard galerkin method do not approximate the solution well using equi-spaced points with a spatial stepsize significantly larger than the sharp boundary layer.
I know that I can overcome this problem by adapting my mesh/nodes to be more densely near the boundary layers, but this presupposes that I know exactly where these layers are. Is there a more generalized approach to this problem which can yield good approximations without apriori knowledge. Ideally, I'm hoping there is a method that allows for a uniformly spaced grid that can get a better approximation than the naive implementations of FDM, FVM, and FEM, without making the mesh extremely fine.
I'm not sure if this is possible, but if anyone knows of any alternative methods that I can try, I'm open to suggestions! :)