On <inline-formula><tex-math id="M1">$ P_1 $</tex-math></inline-formula> nonconforming finite element aproximation for the Signorini problem

Abstract

The main aim of this paper is to study the P 1 nonconforming finite element approximations of the variational inequality arisen from the Signorini problem. We describe the finite dimensional closed convex cone approximation in a meanvalue-oriented sense. In this way, the optimal convergence rate O(h) can be obtained by a refined analysis when the exact solution belongs to H 2 (Ω) without any assumption. Furthermore, we also study the optimal convergence for the case u ∈ H 1+ν (Ω) with 1 2 < ν < 1.