Error Analysis Of The Finite Volume Element Method For Elliptic And Parabolic Partial Differential Equations 1996

Abstract:  An a priori error analysis of the finite volume element method, a locally conservative, Petrov-Galerkin, finite element method for the numerical solution of elliptic and parabolic partial differential equations arising in fluid dynamics applications, is presented. Existing error estimates apply to discretizations of steady diffusion equations by linear finite elements in two spatial dimensions. These results are extended to steady advection-reaction-d iffusion equations and are generalized to polynomial finite elements of arbitrary order in three spatial dimensions and to the full range of admissible regularities for the exact solution. Optimal-order error estimates for h, p, and h-p versions of the method with uniform refinement are derived in a discrete H 1 norm, under minimal regularity assumptions for the exact solution, the finite element triangulation, and the finite volume construction. With additional symmetry assumptions for the finite volumes, multi-dimensional H 1 super...