Duality-based error control for the Signorini problem. (English) Zbl 07892758
Summary: In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in \(\mathrm{L}^p\), for \(p\in (4,\infty)\) in two spatial dimensions. This new analysis treats the positive and negative parts of the discretization error separately, requiring a novel sign- and bound-preserving interpolant, which is shown to have optimal approximation properties. The estimates rely on the sharp dual stability results on the problem in \(\mathrm{W}^{2,(4-\varepsilon)/3}\) for any \(\varepsilon\ll 1\). We summarize extensive numerical experiments aimed at testing the robustness of the estimator to validate the theory.
MSC:
| 35J86 | Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
Keywords:
elliptic variational inequality; a posteriori error control; adaptive meshes; duality methodsReferences:
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