Superclosenes error estimates for the div least-squares finite element method on elliptic problems
Abstract
In this paper we provide some error estimates for the div least-squares finite element method on elliptic problems. The main contribution is presenting a complete error analysis, which improves the current \emph{state-of-the-art} results. The error estimates for both the scalar and the flux variables are established by specially designed dual arguments with the help of two projections: elliptic projection and H(div) projection, which are crucial to supercloseness estimates. In most cases, regularity is omitted to get the optimal convergence rate for vector and scalar unknowns, and most of our results require a lower regularity for the vector variable than the scalar. Moreover, a series of supercloseness results are proved, which are \emph{never seen} in the previous work of least-squares finite element methods.
Cite
@article{arxiv.2404.04918,
title = {Superclosenes error estimates for the div least-squares finite element method on elliptic problems},
author = {Gang Chen and Fanyi Yang and Zheyuan Zhang},
journal= {arXiv preprint arXiv:2404.04918},
year = {2025}
}