Adaptive finite elements for variational inequalities with non-smooth coefficients.

Frédéric Hecht (1, 2) , Zakaria Belhachmi (3)

1 ALPINES - Algorithms and parallel tools for integrated numerical simulations
2 LJLL - Laboratoire Jacques-Louis Lions
3 LMIA - Laboratoire de Mathématiques Informatique et Applications [UHA]

We consider an elliptic variational inequality with discontinuous coefficients arising in unilateral contact mechanics in linearized elasticity. The contact zone is an internal boundary separating sub-domains with difference elastic properties. We study some discrete formulations with mixed finite element methods and we give optimal error estimates in appropriate norms, independent of the variation of the elasticity coefficients. The focus of the article is the a posteriori analysis with residual error indicators. It is achieved both for the conforming and nonconforming discretization, in a unified framework. The residual error indicators are well suited to handle non-matching meshes and the contact conditions, and they allow us to obtain sharp and robust a posteriori estimates. An adaptive solution algorithm is proposed and few numerical experiments confirming the theory are presented.

Nous considérons des discrétisations par éléments finis, conformes et non-conformes pour résoudre le problème de Signorini en contact unilatéral. Nous effectuons l'analyse a postériori avec des indicateurs d'erreur par résidu adaptés aux inéquations variationnelles. Nous obtenons des estimateurs optimaux. Un algorithme numérique mettant en oeuvre l'adaption de maillages avec ces indicateurs est proposé et quelques expériences numériques sont faites.

Domaines: Analyse numérique [cs.NA]

Frédéric Hecht, Zakaria Belhachmi. Adaptive finite elements for variational inequalities with non-smooth coefficients.. 2014. ⟨hal-01011769⟩