Least-Squares versus Partial Least-Squares Finite Element Methods: Robust A Priori and A Posteriori Error Estimates of Augmented Mixed Finite Element Methods
Abstract
In this paper, for the generalized Darcy problem (an elliptic equation with discontinuous coefficients), we study a special partial Least-Squares (Galerkin-least-squares) method, known as the augmented mixed finite element method, and its relationship to the standard least-squares finite element method (LSFEM). Two versions of augmented mixed finite element methods are proposed in the paper with robust a priori and a posteriori error estimates. Augmented mixed finite element methods and the standard LSFEM uses the same a posteriori error estimator: the evaluations of numerical solutions at the corresponding least-squares functionals. As partial least-squares methods, the augmented mixed finite element methods are more flexible than the original LSFEMs. As comparisons, we discuss the mild non-robustness of a priori and a posteriori error estimates of the original LSFEMs. A special case that the -based LSFEM is robust is also presented for the first time. Extensive numerical experiments are presented to verify our findings.
Keywords
Cite
@article{arxiv.2312.15459,
title = {Least-Squares versus Partial Least-Squares Finite Element Methods: Robust A Priori and A Posteriori Error Estimates of Augmented Mixed Finite Element Methods},
author = {Yuxiang Liang and Shun Zhang},
journal= {arXiv preprint arXiv:2312.15459},
year = {2023}
}