Flux-conserving finite element methods

TL;DR

This paper introduces two post-processing methods for finite element solutions that ensure local flux conservation while maintaining optimal order accuracy, verified through numerical tests in 2D and 3D.

Contribution

It proposes novel post-processing techniques that produce flux-conserving finite element solutions of any order without solving non-symmetric equations.

Findings

Finite element solutions approximate flux locally at optimal order.

The proposed methods produce flux-conserving solutions of the same order as original solutions.

Numerical tests confirm the effectiveness of the methods in 2D and 3D.

Abstract

We analyze the flux conservation property of the finite element method. It is shown that the finite element solution does approximate the flux locally in the optimal order, i.e., the same order as that of the nodal interpolation operator. We propose two methods, post-processing the finite element solutions locally. The new solutions, remaining as optimal-order solutions, are flux-conserving elementwise. In one of our methods, the processed solution also satisfies the original finite element equations. While the high-order finite volume schemes are still under construction, our methods produce finite-volume-like finite element solution of any order. In particular, our methods avoid solving non-symmetric finite volume equations. Numerical tests in 2D and 3D verify our findings.