Finite element error estimates in non-energy norms for the two-dimensional scalar Signorini problem

Abstract

This paper is concerned with error estimates for the piecewise linear finite element approximation of the two-dimensional scalar Signorini problem on a convex polygonal domain varOmega . Using a Céa-type lemma, a supercloseness result, and a non-standard duality argument, we prove W^{1,p}(varOmega )-, L^infty (varOmega )-, W^{1,infty }(varOmega )-, and H^{1/2}(partial varOmega )-error estimates under reasonable assumptions on the regularity of the exact solution and L^p(varOmega )-error estimates under comparatively mild assumptions on the involved contact sets. The obtained orders of convergence turn out to be optimal for problems with essentially bounded right-hand sides. Our results are accompanied by numerical experiments which confirm the theoretical findings.