Optimal convergence rates in $L^2$ for a first order system least squares finite element method -- Part II: inhomogeneous Robin boundary conditions

TL;DR

This paper establishes optimal $L^2$ convergence rates for high order finite element discretizations of a first order system least squares method applied to elliptic equations with Robin boundary conditions, supported by numerical validation.

Contribution

It provides the first rigorous analysis of optimal $L^2$ convergence rates for inhomogeneous Robin boundary conditions in this context.

Findings

Optimal $L^2$ convergence rates for scalar variables.

Convergence rates for gradients and vector variables.

Numerical examples confirming theoretical results.

Abstract

We consider divergence-based high order discretizations of an $L^2$-based first order system least squares formulation of a second order elliptic equation with Robin boundary conditions. For smooth geometries, we show optimal convergence rates in the $L^2(\Omega)$ norm for the scalar variable. Convergence rates for the $L^2(\Omega)$-norm error of the gradient of the scalar variable as well as vectorial variable are also derived. Numerical examples illustrate the analysis.