Discrete reliability for high-order Crouzeix--Raviart finite elements

Abstract

In this paper, the adaptive numerical solution of a 2D Poisson model problem by Crouzeix-Raviart elements ($\operatorname*{CR}_{k}$ $\operatorname*{FEM}$) of arbitrary odd degree $k\geq1$ is investigated. The analysis is based on an established, abstract theoretical framework: the \textit{axioms of adaptivity} imply optimal convergence rates for the adaptive algorithm induced by a residual-type a posteriori error estimator. Here, we introduce the error estimator for the $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ discretization and our main theoretical result is the proof ot Axiom 3: \textit{discrete reliability}. This generalizes results for adaptive lowest order $\operatorname*{CR}_{1}$ $\operatorname*{FEM}$ in the literature. For this analysis, we introduce and analyze new local quasi-interpolation operators for $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ which are key for our proof of discrete reliability. We present the results of numerical experiments for the adaptive version of $\operatorname*{CR}_{k}$ $\operatorname*{FEM}$ for some low and higher (odd) degrees $k\geq1$ which illustrate the optimal convergence rates for all considered values of $k$.

Figures (7)

• Figure 1: For the adjacent triangles $K, K^{\prime}$ the purple line depicts the internal boundary $\Gamma$. For $\mathcal{S} := \{K,K^{\prime}\}$, the areas $\omega_{\mathcal{S}^{1/2}}$ and $\omega_{\mathcal{S}^1}$ are encircled by the olive and red lines respectively.
• Figure 2: We set $\mathcal{S} := \{ K, K^{\prime} \}$. Then the purple line denotes the internal boundary $\Gamma$. The set $\mathcal{S}^{1/2}$ is outlined by the olive line. Then $\operatorname{supp} \psi_{\mathcal{S},k}^{\mathbf{z}}$ is encompassed by the dotted line.
• Figure 3: The three ways a triangle $K \in \mathcal{S}^{1/2} \setminus \mathcal{S}$ with $K \cap \partial \Omega = \emptyset$ intersects the interface $\Gamma$ (dashed line).
• Figure 4: The four different ways a triangle $K \in \mathcal{S}^{1/2} \setminus \mathcal{S}$ can interact with the boundary. Here, the dashed line represents the interface $\Gamma$ between $\mathcal{S}$ and $\mathcal{T} \setminus \mathcal{S}$. The hached line segments represent the domain boundary $\partial \Omega$.
• Figure 5: (a): We consider a coarse mesh $\mathcal{T}$ with a triangle $K$ and its patch $\mathcal{T}_K$ marked by the purple line. (b): We consider a refinement $\widehat{\mathcal{T}}$ of the coarse mesh, with the newly added edges marked by the $- \cdot \cdot$ lines. A simplex $\widehat{K} \in \operatorname{succ} ( K )$ is chosen and its fine simplex patch $\widehat{\mathcal{T}}_{\widehat{K}}$ is included by the olive line.

Theorems & Definitions (59)

• Definition 2.2
• Remark 2.3
• Remark 2.5
• Proposition 3.1: CFPP-AxiomsOfAdaptivity
• Theorem 3.2: Discrete reliability
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7