Construction of Locally Conservative Fluxes for High Order Continuous Galerkin Finite Element Methods

TL;DR

This paper introduces a simple post-processing method for high order continuous Galerkin finite element methods to produce locally conservative fluxes with continuous normal components, improving flux accuracy and conservation.

Contribution

The paper presents a novel post-processing technique on control volumes that ensures locally conservative fluxes with continuous normal components for high order CGFEMs.

Findings

Post-processed fluxes converge optimally in H1 semi-norm.

Method maintains low computational cost for any order CGFEM.

Numerical examples confirm the effectiveness of the technique.

Abstract

We propose a simple post-processing technique for linear and high order continuous Galerkin Finite Element Methods (CGFEMs) to obtain locally conservative flux field. The post-processing technique requires solving an auxiliary problem on each element independently which results in solving a linear algebra system whose size is low for any order CGFEM. The post-processing could have been done directly from the finite element solution that results in locally conservative flux on the element. However, the normal flux is not continuous at the elemental boundary. To construct locally conservative flux field whose normal component is also continuous, we propose to do the post-processing on the nodal-centered control volumes which are constructed from the original finite element mesh. We show that the post-processed solution converges in an optimal fashion to the true solution in an H1 semi-norm. We present various numerical examples to demonstrate the performance of the post-processing technique.