Optimal convergence rates in $L^2$ for a first order system least squares finite element method. Part I: homogeneous boundary conditions

TL;DR

This paper proves optimal convergence rates in the L^2 norm for a first order system least squares finite element method applied to elliptic problems with homogeneous boundary conditions, supported by numerical validation.

Contribution

It establishes the first rigorous proof of optimal L^2 convergence for this class of least squares methods on elliptic problems.

Findings

Optimal L^2 convergence rates are achieved.

Numerical results confirm theoretical predictions.

Abstract

We analyze a divergence based first order system least squares method applied to a second order elliptic model problem with homogeneous boundary conditions. We prove optimal convergence in the $L^2(\Omega)$ norm for the scalar variable. Numerical results confirm our findings.