Error estimates in 𝐿², 𝐻¹ and 𝐿^{∞} in covolume methods for elliptic and parabolic problems: A unified approach

Abstract

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 H 1 , L 2 H^1, L^2 norms and new results in the max-norm. For the elliptic problems we demonstrate that the error u − u h u-u_h between the exact solution u u and the approximate solution u h u_h in the maximum norm is O ( h 2 | ln ⁡ h | ) O(h^2|\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.