On $p$-robust convergence and optimality of adaptive FEM driven by equilibrated-flux estimators

Abstract

Building on existing $hp$-adaptive algorithms driven by equilibrated-flux estimators from [ESAIM Math. Model. Numer. Anal. 57 (2023), 329--366] and the references therein, we propose a novel $h$-adaptive algorithm for a fixed polynomial degree $p$. We consider a conforming finite element discretization of the Poisson equation in two or three space dimensions. Supposing piecewise polynomial right-hand side, we show that the algorithm yields error contraction at each step, with a contraction factor that is independent of $p$ provided that a certain {\sl a posteriori} verifiable criterion is satisfied. We further show that this algorithm converges at optimal algebraic rate $s$ if the Dörfler marking parameter is chosen below some specified $p$-independent upper threshold. The constants involved here are $p$-robust, although they may depend on the rate $s$. The theoretical results are supported by numerical experiments, in which the {\sl a posteriori} criterion is always satisfied for one or a few local mesh refinement steps by newest-vertex bisection.

Figures (8)

• Figure 4.1: Example 1. Energy error and a posteriori error estimator as a function of DoFs for polynomial degrees $p\in\{1,2,3,4\}$.
• Figure 4.2: Example 1. Effectivity index of estimator $\eta_\ell(\mathcal{T}_\ell)$ as a function of DoFs for polynomial degrees $p\in\{1,2,3,4\}$ (left). Effectivity index as a function of polynomial degrees $p\in\{1, \dots,13\}$ on a mesh composed of 12 and 48 triangles (right).
• Figure 4.3: Example 1. Maximal and minimal values over all marked vertices $\boldsymbol{a}\in\mathcal{M}_\ell$ of the local stability constants $C_{\text{\rmlb}}(\boldsymbol{a})$ from \ref{['eq:Clb_1']} for polynomial degrees $p\in \{1,2,3,4\}$.
• Figure 4.4: Example 1. Effectivity index of error reduction factor $q_{\rm ctr}$ as a function of DoFs for polynomial degrees $p\in\{1,2,3,4\}$.
• Figure 4.5: Example 2. A posteriori error estimator as a function of DoFs for polynomial degrees $p\in\{1,2,3,4\}$.

Theorems & Definitions (23)

• Theorem 2.1: reliability with constant one ps47
• proof
• Remark 2.2: local equivalence
• Theorem 2.3: local $p$-robust efficiency bps09ev15ev20
• proof
• Lemma 2.4: local discrete efficiency dv23
• proof
• Lemma 2.5: $p$-robust discrete reliability
• proof
• Remark 3.2: choice of $C_{\text{\rmlb,max}}$