First-order system least-squares finite element method for singularly perturbed Darcy equations

TL;DR

This paper introduces a least-squares finite element method for a reformulated Darcy equation that effectively handles singular perturbations, providing reliable error estimation and a straightforward pressure recovery.

Contribution

It presents a novel least-squares approach with a pseudostress variable for singularly perturbed Darcy equations, ensuring uniform stability and efficient error estimation.

Findings

The method is uniformly stable regardless of the perturbation parameter.

The least-squares functional serves as an effective a posteriori error estimator.

Numerical experiments confirm the theoretical results.

Abstract

We define and analyse a least-squares finite element method for a first-order reformulation of a scaled Brinkman model of fluid flow through porous media. We introduce a pseudostress variable that allows to eliminate the pressure variable from the system. It can be recovered by a simple post-processing. It is shown that the least-squares functional is uniformly equivalent, i.e., independent of the singular perturbation parameter, to a parameter dependent norm. This norm equivalence implies that the least-squares functional evaluated in the discrete solution provides an efficient and reliable a posteriori error estimator. Numerical experiments are presented.