Further results on a space-time FOSLS formulation of parabolic PDEs

TL;DR

This paper extends a space-time least-squares finite element approach for parabolic PDEs, proving well-posedness for general second order equations and demonstrating convergence of an adaptive method.

Contribution

It generalizes previous results to broader parabolic PDEs with inhomogeneous boundary conditions and proves convergence of adaptive algorithms.

Findings

Well-posedness for general second order parabolic PDEs

Convergence of adaptive finite element method

Extension of least-squares formulations

Abstract

In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by F\"uhrer& Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven. In the present work, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated. The proof of the latter easily extends to a large class of least-squares formulations.