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

Abstract

AbstractThis 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 $$\varOmega $$ Ω . Using a Céa-type lemma, a supercloseness result, and a non-standard duality argument, we prove $$W^{1,p}(\varOmega )$$ W 1 , p ( Ω ) -, $$L^\infty (\varOmega )$$ L ∞ ( Ω ) -, $$W^{1,\infty }(\varOmega )$$ W 1 , ∞ ( Ω ) -, and $$H^{1/2}(\partial \varOmega )$$ H 1 / 2 ( ∂ Ω ) -error estimates under reasonable assumptions on the regularity of the exact solution and $$L^p(\varOmega )$$ L p ( Ω ) -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.