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 $$\varOmega $$ 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.

Regularity of the Solution of the Scalar Signorini Problem in Polygonal Domains

“…. , m the leading singularity is r 3/2 i since the term r 1/2 i is not in H 2 (Ω), compare the result in Lemma 3.1, and see the discussion in e. g. [3,14]. We show now y r = o(r π/(2α i ) i…”

“…Example 2.6. Christof and Haubner investigated in [3] a square domain and the case Γ = Γ S . In the case of a homogeneous differential equation, Condition (A) in this paper is now proven in Theorem 2.4, namely that the relative boundary of Γ C has onedimensional Hausdorff measure zero and the relative interior of Γ C consists of at most finitely many connected components.…”

“…We formulate now a regularity result in the spirit of Theorem 2.3 of the paper [3] by Christof and Haubner where the regular part of the solution is considered in W 2,p , p > 2, p = 4. But we like to note that the regular part could also be smoother; the prize is that possibly more singular terms have to be included and the datum u at the Neumann boundary must be sufficiently regular.…”

“…Otherwise the set of endpoints of the coincidence set (the set of points where the condition u = 0 changes to u > 0) could possess accumulation points while the analysis of the regularity near such points (or near corners of the domain) assumes the existence of a δ-neighborhood where the type of the boundary condition does not change. As a consequence there are publications where the structure of the coincidence set is formulated as an assumption, see, e.g., [3,Condition (A)].…”

Error analysis of the compliance model for the Signorini problem

“…We mention the early works in [27] used in [10] where a complicated adjoint problem is involved and whose regularity is not established. Using linear finite elements, we also refer to the recent works [9] where an L 4 -estimate of order 2 − ε is obtained in two dimensions with additional assumptions on the contact sets, and [30] where authors prove a 3/2 − ε error estimate in the L 2 -norm when the solution is H 5/2−ε -regular. Besides, the numerous numerical results dealing with L 2 -error estimates (see, e.g., [9,20,21,30]) indicate that the convergences behave well and are optimal, so the main lack comes from the mathematical analysis.…”