Suppose I have a PDE in a rectangular domain that I am solving numerically via a finite difference method. How do I answer the question, "How fine do I need to make the grid to so that my solution is within $\epsilon$ of the true solution?" We can assume some basic interpolation scheme is used to fill in the points outside of the grid.
For a concrete example, we might consider solving a 2-d heat equation with Crank-Nicolson. But I'd like methods that apply more broadly.
My understanding is that for finite element methods, there is a well-developed theory of "a posteriori" estimates that answer my question. Does a theory providing rigorous error bounds exist for finite difference methods?