Error Estimates in L, H and L∞ in Covolume Methods for Elliptic and Parabolic Problems: a Unified Approach

In this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane. We introduce a general approach for connecting these methods with finite element method analysis. This unified approach is used to prove known convergence results in the H1, L2 norms and new results in the max-norm. For the elliptic problems we demonstrate that the error u−uh between the exact solution u and the approximate solution uh in the maximum norm is O(h2| ln h|) in the linear element case. Furthermore, the maximum norm error in the gradient is shown to be of first order. Similar results hold for the parabolic problems.