$\textit{A priori}$ and $\textit{a posteriori}$ error identities for the scalar Signorini problem

TL;DR

The paper addresses the scalar Signorini problem, a nonlinear boundary-contact model, and derives both a posteriori and a priori error identities via Fenchel duality. It combines continuous-level duality with discrete CR/RT discretizations to obtain a posteriori error identities and a priori error identities that yield quasi-optimal convergence without imposing extra regularity on the contact set, applicable in arbitrary dimensions. A primal–dual gap framework enables local refinement indicators and a reconstruction strategy for simultaneous primal and dual approximations. The approach enhances reliable error control and adaptive refinement for unilateral contact problems, providing solid convergence guarantees across dimensions and problem settings.

Abstract

In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an $\textit{a posteriori}$ error identity for arbitrary conforming approximations of the primal formulation and the dual formulation of the scalar Signorini problem. In addition, on the basis of a (Fenchel) duality theory on the discrete level, we derive an $\textit{a priori}$ error identity that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element, and leads to quasi-optimal error decay rates without imposing additional assumptions on the contact set and in arbitrary space dimensions.

Figures (5)

• Figure 1: left: $\Omega$, $\Gamma_C$, $\Gamma_D$, $\{u>0\}\cap \Gamma_C=\{z\cdot n=0\}\cap \Gamma_C$, and $\{u=0\}\cap \Gamma_C=\{z\cdot n>0\}\cap \Gamma_C$; right: $\psi, \psi^{\prime},\psi^{\prime\prime}, \psi^{\prime\prime\prime}, \psi^{\prime\prime\prime\prime}\colon [0,1]\to\mathbb{R}$.
• Figure 2: left: discrete primal solution $u_{h_5}^{cr}\in K_{h_5}^{cr}$, where red stars mark sides $S\in \mathcal{T}_{h_5}$ with $\pi_{h_5}u_{h_5}^{cr}|_{S}>0$; right: (local) $L^2$-projection (onto $(\mathcal{L}^0(\mathcal{T}_{h_5}))^d$) of discrete dual solution~ ${z_{h_5}^{rt}\space\in \space K_{h_5}^{rt,*}}$, where red squares mark sides $S\in \mathcal{T}_{h_5}$ with $z_{h_5}^{rt}\cdot n|_{S}>0$. We find that $z_{h_5}^{rt}\cdot n\,\pi_{h_5}u_{h_5}^{cr}=0$ a.e. on $\Gamma_C$.
• Figure 3: Logarithmic plots of the experimental convergence rates of the error quantities \ref{['eq:error_quantities']}. We observe the experimental orders of convergence $\texttt{EOC}_k(e_k^{\textup{tot}})\approx\texttt{EOC}_k(e_k^{\textup{gap}})\approx 2$, $k=1,\ldots, 7$, and $\texttt{EOC}_k(e_k^{\Delta})\approx 3.7$, $k=1,\ldots, 7$.
• Figure 4: left: $\Omega$, $\Gamma_C$, $\Gamma_D$, $\Gamma_N$, and $(1,0)^\top$; right: primal-dual gap estimator ${\eta^2_{\textup{gap}}}(\overline{u}_{h_k}^{cr},z_{h_k}^{rt})$ for $k=0,\dots,20$, when employing adaptive mesh refinement (i.e., $\theta=\frac{1}{2}$ in Algorithm \ref{['alg:afem']}), and for $k=0,\dots, 4$, when employing uniform mesh refinement (i.e., $\theta=1$ in Algorithm \ref{['alg:afem']}).
• Figure 5: left: discrete primal solution $u_{h_{15}}^{cr}\in K_{h_{15}}^{cr}$; MIDDLE: discrete Lagrange multiplier ${\overline{\lambda}}_{h_{15}}^{cr}\in \mathcal{L}^0(\mathcal{S}_{h_{15}}^{\Gamma_{C}})$; right: (local) $L^2$-projection of the discrete dual solution ${z_{h_{15}}^{rt}\in K_{h_{15}}^{rt,*}}$.

Theorems & Definitions (31)

• Proposition 3.1: strong duality and convex duality relations
• Remark 3.2
• proof : Proof (of Proposition \ref{['prop:duality']}).
• Remark 3.3: regularity in 2D
• Lemma 4.1
• Remark 4.2: interpretation of the components of the primal-dual gap estimator
• proof : Proof (of Lemma \ref{['lem:primal_dual_gap_estimator']}).
• Lemma 4.3: optimal strong convexity measures
• Remark 4.4
• proof : Proof (of Lemma \ref{['lem:strong_convexity_measures']})