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

TL;DR

The paper develops and analyzes a high-order hp-FOSLS method for a Poisson-type elliptic problem with inhomogeneous Robin boundary conditions, proving optimal $L^2$ convergence for the scalar variable and for related quantities, by combining duality arguments with a novel constrained projection $\pmb{I}^\Gamma_h$. Key contributions include norm-equivalence results, a commuting-diagram framework, and a carefully constructed $\pmb{I}^\Gamma_h$ that captures both volume and boundary trace orthogonality, together with Helmholtz decompositions to manage regularity limitations. The error analysis proceeds via bootstrapping suboptimal estimates to optimal bounds for $e^u$ and the vector flux, as well as optimal control of boundary traces, under suitable elliptic-regularity shifts. Numerical experiments confirm the theoretical rates and illustrate suboptimalities and superconvergence phenomena in practice, highlighting the method’s robustness for smooth domains and finite-regularity data.

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 $L^2(Ω)$ norm for the scalar variable. Convergence rates for the $L^2(Ω)$-norm error of the gradient of the scalar variable as well as vectorial variable are also derived. Numerical examples illustrate the analysis.

Figures (7)

• Figure 1: $h$-convergence of $\left\|e^u\right\|_{L^2(\Omega)}$ using $\pmb{\mathrm{V}}_{p_v}(\mathcal{T}_h) = \pmb{\mathrm{RT}}_{p_v-1}(\mathcal{T}_h)$, see Example \ref{['example:numerics_singular_solution_robin']}.
• Figure 2: $h$-convergence of $\left\|e^u\right\|_{L^2(\Omega)}$ using $\pmb{\mathrm{V}}_{p_v}(\mathcal{T}_h) = \pmb{\mathrm{BDM}}_{p_v}(\mathcal{T}_h)$, see Example \ref{['example:numerics_singular_solution_robin']}.
• Figure 3: $h$-convergence of $\left\|\nabla e^u\right\|_{L^2(\Omega)}$ using $\pmb{\mathrm{V}}_{p_v}(\mathcal{T}_h) = \pmb{\mathrm{RT}}_{p_v-1}(\mathcal{T}_h)$, see Example \ref{['example:numerics_singular_solution_robin']}.
• Figure 4: $h$-convergence of $\left\|\nabla e^u\right\|_{L^2(\Omega)}$ using $\pmb{\mathrm{V}}_{p_v}(\mathcal{T}_h) = \pmb{\mathrm{BDM}}_{p_v}(\mathcal{T}_h)$, see Example \ref{['example:numerics_singular_solution_robin']}.
• Figure 5: $h$-convergence of $\left\|\pmb{e}^{\pmb{\varphi}}\right\|_{L^2(\Omega)}$ using $\pmb{\mathrm{V}}_{p_v}(\mathcal{T}_h) = \pmb{\mathrm{RT}}_{p_v-1}(\mathcal{T}_h)$, see Example \ref{['example:numerics_singular_solution_robin']}.

Theorems & Definitions (48)

• Remark 1.2
• Remark 2.1
• Theorem 2.2: Norm equivalence - Robin version of bernkopf-melenk22
• proof
• Remark 3.2
• Theorem 3.3: Duality argument for the scalar variable --- Robin version of bernkopf-melenk22
• proof
• Theorem 3.4: Duality argument for the gradient of the scalar variable - Robin version of bernkopf-melenk22
• proof
• Theorem 3.5: Duality argument for the vector valued variable --- Robin version of bernkopf-melenk22