The Finite Volume Element Method for Diffusion Equations on General Triangulations

Abstract

This paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in $\mathcal{R}^2 $ using a special type of control volume. The theory applies to diffusion equations of the form \[ \begin{gathered} - \nabla (A\nabla u) = f\quad {\text{in }}\Omega , \hfill \\ u = 0\quad {\text{on }}\partial \Omega . \hfill \\ \end{gathered} \] Under fairly general conditions, the theory establishes $O(h)$ estimates of the error in a discrete $\mathcal{H}^1 $ seminorm. Under an additional assumption concerning local uniformity of the triangulation, the estimate is improved to $O(h^2 )$.