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

TL;DR

This work develops a divergence-based first order system least squares method for a Poisson-like elliptic model with homogeneous boundary conditions and proves optimal $L^2(\Omega)$ convergence for the scalar variable $u$, using a refined duality framework and hp-FEM analysis. It introduces div-conforming projection operators $\mathcal{I}_h^0$ and $\mathcal{I}_h$ and discrete Helmholtz decompositions to overcome limitations of standard duality estimates, enabling hp-robust error bounds for the gradient $\nabla u$ and the vector variable $\pmb{\varphi}$. The theory is corroborated by numerical experiments on curved elements, comparing Raviart-Thomas and Brezzi-Douglas-Marini spaces and confirming optimal rates when the vector space degree is suitably chosen (e.g., $p_v>1$ and $p_v=p_s-1$). The results provide rigorous L2-error estimates and guidance on selecting vector spaces in practical hp-FEM implementations of FOSLS for second-order elliptic problems.

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 $L^2(Ω)$ norm for the scalar variable. Numerical results confirm our findings.

Figures (12)

• Figure 5.1: (cf. Example \ref{['example:numerics_smooth_solution']}) Convergence of $\left\lVert e^u\right\rVert_{L^2(\Omega)}$ vs. $\sqrt{\operatorname{DOF}} \sim 1/h$ employing $\pmb{\mathrm{V}}^0_{p_v}(\mathcal{T}_h) = \pmb{\mathrm{RT}}^0_{p_v-1}(\mathcal{T}_h)$.
• Figure 5.2: (cf. Example \ref{['example:numerics_smooth_solution']}) Convergence of $\left\lVert\pmb{e}^{\pmb{\varphi}}\right\rVert_{L^2(\Omega)}$ vs. $\sqrt{\operatorname{DOF}} \sim 1/h$ employing $\pmb{\mathrm{V}}^0_{p_v}(\mathcal{T}_h) = \pmb{\mathrm{RT}}^0_{p_v-1}(\mathcal{T}_h)$.
• Figure 5.3: (cf. Example \ref{['example:numerics_smooth_solution']}) Convergence of $\left\lVert\pmb{e}^{\pmb{\varphi}}\right\rVert_{L^2(\Omega)}$ vs. $\sqrt{\operatorname{DOF}} \sim 1/h$ employing $\pmb{\mathrm{V}}^0_{p_v}(\mathcal{T}_h) = \pmb{\mathrm{BDM}}^0_{p_v}(\mathcal{T}_h)$.
• Figure 5.4: (cf. Example \ref{['example:numerics_singular_solution']}) Convergence of $\left\lVert e^u\right\rVert_{L^2(\Omega)}$ vs. $\sqrt{\operatorname{DOF}} \sim 1/h$ employing $\pmb{\mathrm{V}}^0_{p_v}(\mathcal{T}_h) = \pmb{\mathrm{RT}}^0_{p_v-1}(\mathcal{T}_h)$.
• Figure 5.5: (cf. Example \ref{['example:numerics_singular_solution']}) Convergence of $\left\lVert\nabla e^u\right\rVert_{L^2(\Omega)}$ vs. $\sqrt{\operatorname{DOF}} \sim 1/h$ employing $\pmb{\mathrm{V}}^0_{p_v}(\mathcal{T}_h) = \pmb{\mathrm{RT}}^0_{p_v-1}(\mathcal{T}_h)$.

Theorems & Definitions (38)

• Theorem 2.1: Norm equivalence
• proof
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Theorem 3.1: Duality argument for the scalar variable
• proof
• Theorem 3.2: Duality argument for the gradient of the scalar variable
• proof
• Theorem 3.3: Duality argument for the vector valued variable