Duality based error control for the Signorini problem

Ben S. Ashby; Tristan Pryer

doi:10.1137/22M1534791

Duality based error control for the Signorini problem

Ben S. Ashby, Tristan Pryer

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Abstract

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 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 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.

Original language	English
Pages (from-to)	1687-1712
Number of pages	26
Journal	SIAM Journal on Numerical Analysis
Volume	62
Issue number	4
Early online date	23 Jul 2024
DOIs	https://doi.org/10.1137/22M1534791
Publication status	Published - 1 Aug 2024

Funding

The first author was supported through a Ph.D. scholarship awarded by the EPSRC Centre for Doctoral Training in the Mathematics of Planet Earth at Imperial College London and the University of Reading EP/L016613/1. The second author received partial support from the EPSRC programme grant EP/W026899/1. Both authors were supported by the Leverhulme RPG-2021-238. \\ast Received by the editors November 14, 2022; accepted for publication (in revised form) March 29, 2024; published electronically July 23, 2024. https://doi.org/10.1137/22M1534791 Funding: The first author was supported through a Ph.D. scholarship awarded by the EPSRC Centre for Doctoral Training in the Mathematics of Planet Earth at Imperial College London and the University of Reading EP/L016613/1. The second author received partial support from the EPSRC programme grant EP/W026899/1. Both authors were supported by the Leverhulme RPG-2021-238. \\dagger Institute for Mathematical Innovation, University of Bath, BA2 7AY, Bath, UK (bsa34@ bath.ac.uk). \\ddagger Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK (tmp38@ bath.ac.uk).

Funders	Funder number
EPSRC Centre for Doctoral Training
University of Reading	EP/L016613/1
University of Reading
EPSRC - EU	EP/W026899/1
Leverhulme Trust	RPG-2021-238
Leverhulme Trust

Keywords

a posteriori error control
adaptive meshes
duality methods
elliptic variational inequality

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics
Numerical Analysis

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abstract = "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 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 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.",

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AB - 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 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 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.

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Duality based error control for the Signorini problem

Abstract

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Keywords

ASJC Scopus subject areas

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