New a priori analysis of first-order system least-squares finite element methods for parabolic problems

TL;DR

This paper introduces a new a priori analysis for least-squares finite element methods applied to parabolic first-order systems, providing optimal error estimates and confirming results through numerical experiments.

Contribution

It develops a novel elliptic projection operator based on a non-symmetric bilinear form, enabling improved error analysis for these methods.

Findings

Optimal error estimates in natural norms

Enhanced understanding of least-squares FEM for parabolic systems

Numerical results confirm theoretical predictions

Abstract

We provide new insights into the a priori theory for a time-stepping scheme based on least-squares finite element methods for parabolic first-order systems. The elliptic part of the problem is of general reaction-convection-diffusion type. The new ingredient in the analysis is an elliptic projection operator defined via a non-symmetric bilinear form, although the main bilinear form corresponding to the least-squares functional is symmetric. This new operator allows to prove optimal error estimates in the natural norm associated to the problem and, under additional regularity assumptions, in the $L^2$ norm. Numerical experiments are presented which confirm our theoretical findings.