Construction of locally conservative fluxes for high order continuous Galerkin finite element methods

Abstract

Despite their robustness, it is known that standard continuous Galerkin Finite Element Methods (CGFEMs) do not produce a locally conservative flux field. As a result, their application to solving model problems that are derived from conservation laws can be limited. To remedy this issue some form of post-processing must be performed on the CGFEM solution. In this work, a simple post-processing technique is proposed to obtain a locally conservative flux field from a CGFEM solution. One distinct advantage of the proposed method is that it produces continuous normal flux at the element’s boundary. The post-processing is implemented on nodal-centered control volumes that are constructed from the original finite element mesh. The post-processing method is performed by solving an independent set of low dimensional problems posed on each element. The associated linear algebra systems are of dimension 1 2 ( k + 1 ) ( k + 2 ) where k is the polynomial degree of CGFEM basis on a triangular mesh. A theoretical investigation is conducted to confirm that the post-processed solution converges in an optimal fashion to the true solution in the H 1 semi-norm. Various numerical examples that demonstrate the performance of technique are given. Specifically, a simulation of a model for single-phase flow in a heterogeneous system is presented to show the necessity of the local conservation as well as the effective performance of the post-processing technique.