Adaptive Least-Squares Finite Element Methods for Linear Transport Equations Based on an H(div) Flux Reformulation

TL;DR

This paper introduces adaptive least-squares finite element methods for linear transport equations that effectively handle discontinuities and boundary conditions by reformulating the flux variable in an $H( ext{div})$ space, improving accuracy and error estimation.

Contribution

The paper proposes a novel flux reformulation in $H( ext{div})$ spaces for LSFEMs, enabling better handling of discontinuities and boundary conditions in linear transport equations.

Findings

New LSFEMs handle discontinuous solutions more effectively.

Adaptive methods identify error sources like singularities.

Almost no overshoot with $RT_0 imes P_0$ approximation on discontinuous solutions.

Abstract

In this paper, we study the least-squares finite element methods (LSFEM) for the linear hyperbolic transport equations. The linear transport equation naturally allows discontinuous solutions and discontinuous inflow conditions, while the normal component of the flux across the mesh faces needs to be continuous. Traditional LSFEMs using continuous finite element approximations will introduce unnecessary extra error for discontinuous solutions and boundary conditions. In order to separate the continuity requirements, a new flux variable is introduced. With this reformulation, the continuities of the flux and the solution can be handled separately in natural $H(\mbox{div};\Omega)\times L^2(\Omega)$ conforming finite element spaces. Several variants of the methods are developed to handle the inflow boundary condition strongly or weakly. With the reformulation, the new LSFEMs can handle discontinuous solutions and boundary conditions much better than the traditional LSFEMs with continuous polynomial approximations. With least-squares functionals as a posteriori error estimators, the adaptive methods can naturally identify error sources including singularity and non-matching discontinuity. For discontinuity aligned mesh, no extra error is introduced. If an $RT_0 \times P_0$ pair is used to approximate the flux and the solution, the new adaptive LSFEMs can approximate discontinuous solutions with almost no overshooting even when the mesh is not aligned with discontinuity. Existence and uniqueness of the solutions and a priori and a posteriori error estimates are established for the proposed methods. Extensive numerical tests are performed to show the effectiveness of the methods developed in the paper.