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

TL;DR

This paper develops new a priori and a posteriori error identities for the scalar Signorini problem using duality theory, enabling accurate error estimation for conforming approximations in arbitrary dimensions.

Contribution

It introduces novel error identities based on duality theory for both continuous and discrete levels, applicable to primal and dual formulations of the Signorini problem.

Findings

Derives an a posteriori error identity for conforming approximations.

Establishes an a priori error identity with quasi-optimal decay rates.

Applicable to arbitrary space dimensions without extra contact set assumptions.

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.