Suppose I want to solve a PDE of the form
\begin{align} Lu &= f, \quad \mathrm{in}\; \Omega \\[1em] u &= g_D, \quad \mathrm{on}\; \partial\Omega_D \\[1em] u &= g_N, \quad \mathrm{on}\; \partial\Omega_N \end{align} where $u$ has a singularity somewhere in $\Omega$. Since finite difference methods are based on approximating derivatives of a given function at grid points, it would seem to be the case that smoothness of $u$ is a necessary condition for any FD scheme to converge. Is it possible to cook up a variation of the finite difference method in which the smoothness requirement is relaxed? Or is this simply a fundamental limitation of FD methods?
I have in mind a BVP in which $f$ has a singularity - if $f$ was replaced by some $\tilde{f} \in C^{\infty}(\Omega)$ and this new, smooth rhs function was evaluated at the FD grid points, would one expect the resulting finite difference scheme to converge?