A Priori and A Posteriori Error Identities for the Scalar Signorini Problem

Abstract

In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an 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 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.