Review and computational comparison of adaptive least-squares finite element schemes

TL;DR

This paper reviews convergence results of adaptive least-squares finite element methods and empirically compares three mesh-refinement strategies on benchmark elliptic PDE problems, highlighting parameter choices and data approximation.

Contribution

It provides a comprehensive review of convergence analyses and offers a computational comparison of adaptive strategies for least-squares finite element methods.

Findings

Different adaptive strategies show varying efficiency depending on parameters.

Data approximation significantly impacts the convergence behavior.

Numerical experiments are reproducible with the author's software package.

Abstract

The convergence analysis for least-squares finite element methods led to various adaptive mesh-refinement strategies: Collective marking algorithms driven by the built-in a posteriori error estimator or an alternative explicit residual-based error estimator as well as a separate marking strategy based on the alternative error estimator and an optimal data approximation algorithm. This paper reviews and discusses available convergence results. In addition, all three strategies are investigated empirically for a set of benchmarks examples of second-order elliptic partial differential equations in two spatial dimensions. Particular interest is on the choice of the marking and refinement parameters and the approximation of the given data. The numerical experiments are reproducible using the author's software package octAFEM available on the platform Code Ocean.