Consider the 2-dimensional Helmholtz equation
$$-\Delta u-k^2 u= f.$$
I'm trying to proof convergence of the standard finite difference scheme (using the 5-point-stencil for the Laplacian). I have seen some proofs for the similar looking equation
$$-\Delta u+ cu= f$$
with $c\geq 0$. But since those proofs usually use some version of a maximum principle (which doesn't hold true for the Helmholtz equation), I'm stuck. Does anyone have a reference for a proof where $c\geq 0$ isn't required?