Duality based error control for the Signorini problem

In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in Lᵖ, for p ∈ (4,∞) in two spatial dimensions. This new analysis treats the positive and negative parts of the discretisation error separately, requiring a novel sign- and bound-preserving interpolant, which is shown to have optimal approximation properties. The estimates rely on the sharp dual stability results on the problem in W2,(4 - ε)/3 for any ε 1. We summarise extensive numerical experiments aimed at testing the robustness of the estimator to validate the theory.