Adaptive Least-Squares Finite Element Methods for Linear Transport Equations Based on Flux Reformulation

Abstract

In this thesis, several least-squares finite element methods(LSFEMs) are developed for solving 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. Compared to elliptic equation, the solution space has no conforming finite element subspace. Traditional LSFEMs using continuous finite element approximations will introduce unnecessary extra error for discontinuous solutions and boundary conditions, Gibbs phenomenon is very possible near the discontinuity. In order to separate the continuity requirements, a new flux variable is introduced. After this reformulation, the continuities of the flux and the solution can be handled separately in natural H (div; Ω) x L²(Ω) conforming finite element spaces. Several variants of the methods are developed to handle the inflow boundary condition strongly or weakly.

With the reformulation, the proposed methods can use the lowest order finite element approximation spaces: RT₀ and P₀. Using such lowest order element combined with natural least-square functionals as a posteriori error estimators, the methods can resolve the discontinuity even when the mesh is not aligned with discontinuity. The smearing and overshooting phenomena are also very mild with adaptive methods. In this way, the least-squares finite element methods can handle discontinuous solutions much better than the traditional continuous polynomial approximations.

Realizing that the Raviart-Thomas mixed element space has enough degrees of freedom to approximate both the flux and its divergence, a further improvement is to eliminate the solution from the system and develop LSFEMs for new flux-only system. The solution then is recovered by simple post-processing methods using its relation with the flux. This kind of flux-only LSFEMs use less DOFs than the original try with both flux and solution. 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.

The existence, uniqueness of the solution, a priori and a posteriori error estimates are also analyzed for the proposed methods. Extensive numerical tests are done to show the effectiveness of the proposed methods in this thesis.

Date of Award	21 Apr 2020
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Shun ZHANG (Supervisor)