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