Div First-Order System LL* (FOSLL*) for Second-Order Elliptic Partial Differential Equations

TL;DR

This paper applies the FOSLL* approach to second-order elliptic PDEs, establishing well-posedness and error bounds, and introduces an explicit residual error estimator for finite element approximations.

Contribution

It extends the FOSLL* method to the div system, proves its well-posedness, and develops a reliable and efficient residual error estimator.

Findings

FOSLL* approach is well-posed for div systems.

Finite element approximation achieves quasi-optimal error bounds.

An explicit residual error estimator is proposed with proven reliability and efficiency.

Abstract

The first-order system LL* (FOSLL*) approach for general second-order elliptic partial differential equations was proposed and analyzed in [10], in order to retain the full efficiency of the L2 norm first-order system least-squares (FOSLS) ap- proach while exhibiting the generality of the inverse-norm FOSLS approach. The FOSLL* approach in [10] was applied to the div-curl system with added slack vari- ables, and hence it is quite complicated. In this paper, we apply the FOSLL* approach to the div system and establish its well-posedness. For the corresponding finite ele- ment approximation, we obtain a quasi-optimal a priori error bound under the same regularity assumption as the standard Galerkin method, but without the restriction to sufficiently small mesh size. Unlike the FOSLS approach, the FOSLL* approach does not have a free a posteriori error estimator, we then propose an explicit residual error estimator and establish its reliability and efficiency bounds