Error analysis for a Crouzeix-Raviart approximation of the obstacle problem

Abstract

In the present paper, we study a Crouzeix-Raviart approximation of the obstacle problem, which imposes the obstacle constraint in the midpoints (i.e., barycenters) of the elements of a triangulation. We establish a priori error estimates imposing natural regularity assumptions, which are optimal, and the reliability and efficiency of a primal-dual type a posteriori error estimator for general obstacles and involving data oscillation terms stemming only from the right-hand side. Numerical experiments are carried out to support the theoretical findings.

Figures (6)

• Figure 1: left: Plots of $\eta_k^2(v_k)$ and $\tilde{\rho}_k^2(v_k)$ for $v_k\coloneqq \max\{\Pi_{h_k}^{av} u_k^{cr},\chi\}\in K$ using adaptive mesh refinement for $k=0,\dots,20$ and using uniform mesh refinement for $k=0,\dots, 4$. right: Plots of $I(v_k)$ (cf. \ref{['eq:obstacle_primal']}), for $v_k\coloneqq \max\{\Pi_{h_k}^{av} u_k^{cr},\chi\}\in K$ and $D(z_k^{rt})$ (cf. \ref{['eq:obstacle_dual']}), using adaptive mesh refinement for $k=0,\dots,20$ and using uniform mesh refinement for $k=0,\dots, 4$.
• Figure 2: Adaptively refined meshes $\mathcal{T}_k$, $k\in \{0,4,8,12,16,20\}$, with approximate contact zones $\mathcal{C}_k^{cr}\coloneqq\{\Pi_{h_k}u_k^{cr}=0\}=\{{\overline{\lambda}}_k^{cr}<0\}$, $k\in \{0,4,8,12,16,20\}$, shown in white.
• Figure 3: upper left: discrete primal solution $u_{10}^{cr}\in \mathcal{S}^{1,cr}_D(\mathcal{T}_{10})$; upper right: node-averaged discrete primal solution $\Pi_{h_{10}}^{av}u_{10}^{cr}\in \mathcal{S}^1_D(\mathcal{T}_{10})$; lower left: discrete Lagrange multiplier ${\overline{\lambda}}_{10}^{cr}\in \Pi_{h_{10}}(\mathcal{S}^{1,cr}_D(\mathcal{T}_{10}))$; lower right: and discrete dual solution $z_{10}^{rt}\in \mathcal{R}T^0_N(\mathcal{T}_{10})$.
• Figure 4: left: Plots of $\eta_k^2(v_k)$ and $\tilde{\rho}_k^2(v_k)$ for $v_k\coloneqq \max\{\Pi_{h_k}^{av} u_k^{cr},\chi\}\in K$ using adaptive mesh refinement for $k=0,\dots,25$ and using uniform mesh refinement for $k=0,\dots, 4$. right: Plots of $I(v_k)$, cf. \ref{['eq:obstacle_primal']}, for $v_k\coloneqq \max\{\Pi_{h_k}^{av} u_k^{cr},\chi\}\in K$ and $D(z_k^{rt})$, cf. \ref{['eq:obstacle_dual']}, using adaptive mesh refinement for $k=0,\dots,25$ and using uniform mesh refinement for $k=0,\dots, 4$.
• Figure 5: Adaptively refined meshes $\mathcal{T}_k$, $k\in\{0,5,10,15,20,25\}$, with approximate contact zones $\mathcal{C}_k^{cr}\coloneqq\{\Pi_{h_k}u_k^{cr}=\Pi_{h_k}\chi_{h_k}\}=\{{\overline{\lambda}}_k^{cr}<0\}$, $k\in\{0,5,10,15,20,25\}$, shown in white.

Theorems & Definitions (43)

• Proposition 3.1
• Remark 3.2
• proof : Proof (of Proposition \ref{['prop:augmented']}).
• Proposition 3.3
• proof
• Theorem 4.1
• proof
• Corollary 4.2
• proof
• Remark 5.1