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

Abstract

<p style='text-indent:20px;'>The main aim of this paper is to study the <inline-formula><tex-math id="M2">$ P_1 $</tex-math></inline-formula> nonconforming finite element approximations of the variational inequality arisen from the Signorini problem. We describe the finite dimensional closed convex cone approximation in a meanvalue-oriented sense. In this way, the optimal convergence rate <inline-formula><tex-math id="M3">$ O(h) $</tex-math></inline-formula> can be obtained by a refined analysis when the exact solution belongs to <inline-formula><tex-math id="M4">$ H^{2}(\Omega) $</tex-math></inline-formula> without any assumption. Furthermore, we also study the optimal convergence for the case <inline-formula><tex-math id="M5">$ u\in H^{1+\nu}(\Omega) $</tex-math></inline-formula> with <inline-formula><tex-math id="M6">$ \frac{1}{2}<\nu<1 $</tex-math></inline-formula>.</p>