Adaptive Flux-Only Least-Squares Finite Element Methods for Linear Transport Equations

TL;DR

This paper introduces two flux-only least-squares finite element methods for linear transport equations that effectively handle discontinuities and improve accuracy by eliminating the solution variable and focusing on flux approximation.

Contribution

The paper develops two novel flux-only LSFEMs that use fewer degrees of freedom and better manage discontinuities compared to previous methods, with proven error estimates and adaptive refinement.

Findings

Methods handle discontinuous solutions effectively

Adaptive mesh refinement improves accuracy near discontinuities

Numerical tests confirm the effectiveness of the proposed methods

Abstract

In this paper, two flux-only least-squares finite element methods (LSFEM) for the linear hyperbolic transport problem are developed. The transport equation often has discontinuous solutions and discontinuous inflow boundary conditions, but the normal component of the flux across the mesh interfaces is continuous. In traditional LSFEMs, the continuous finite element space is used to approximate the solution. This will cause unnecessary error around the discontinuity and serious overshooting. In arXiv:1807.01524 [math.NA], we reformulate the equation by introducing a new flux variable to separate the continuity requirements of the flux and the solution. Realizing that the Raviart-Thomas mixed element space has enough degrees of freedom to approximate both the flux and its divergence, we eliminate the solution from the system and get two flux-only formulations, and develop corresponding LSFEMs. The solution then is recovered by simple post-processing methods using its relation with the flux. These two versions of flux-only LSFEMs use less DOFs than the method we developed in arXiv:1807.01524 [math.NA]. Similar to the LSFEM developed in arXiv:1807.01524 [math.NA], both flux-only LSFEMs can handle discontinuous solutions better than the traditional continuous polynomial approximations. We show the existence, uniqueness, a priori and a posteriori error estimates of the proposed methods. With adaptive mesh refinements driven by the least-squares a posteriori error estimators, the solution can be accurately approximated even when the mesh is not aligned with discontinuity. The overshooting phenomenon is very mild if a piecewise constant reconstruction of the solution is used. Extensive numerical tests are done to show the effectiveness of the methods developed in the paper.