The Finite Volume Element Method for Diffusion Equations on General Triangulations

This paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in R² using a special type of control volume. The theory applies to diffusion equations of the form \[ {gathered} - ∇ (A∇ u) = f {in }Ω , \\ u = 0 {on }∂ Ω . \\ {gathered} \] Under fairly general conditions, the theory establishes $O(h)$ estimates of the error in a discrete H¹ seminorm. Under an additional assumption concerning local uniformity of the triangulation, the estimate is improved to O(h² ).