An Approach to Improve the Convergence Order in Finite Volume and Finite Element Methods

Abstract

It is well known that to obtain a highly accurate numrical approximations in finite element and in finite difference methods, we use in general finite element spaces of high degree, in case of finite element methods, and schemes of higher order in case of finite difference methods. The disadvantage of these techniques is that they lead to complex linear systems to be solved. We are interested to use lower order schemes, instead of higher order schemes, to produce higher order numerical approximations. several approaches have been devoted to get higher order approximations using lower order schemes, e.g. defect correction in finite element method, see for instance [29, 16, 5], and deferred and difference correction in finite difference method, see for instance [29, 26]. These previous approaches are limited to the 1d case or to the multi‐d case with “uniform meshes” in finite difference and finite element methods. Consequently, it seems not straightforward to apply previous stated approaches to get higher order approximation using lower order schemes in finite volume methods in which the meshes are unstructured. Nevertheless, we present a new approach to get higher order approximations using lower order schemes in finite volume and finite element methods methods. This approach could be applied even when the mesh is “unstructured.” The aim of this short paper is to present our conceptual approach.