Analysis of a VEM-fully discrete polytopal scheme with bubble stabilisation for contact mechanics with Tresca friction

Analyse d'un schéma polytopal entièrement discret VEM avec stabilisation par bulle pour la mécanique des contacts avec friction de Tresca.

Jérôme Droniou (1) , Ali Haidar (2, 3) , Roland Masson (2)

1 Analyse, Calcul Scientifique Industriel et Optimisation de Montpellier (ACSIOM)
2 COFFEE - COmplex Flows For Energy and Environment
3 IFPEN - IFP Energies nouvelles

This work performs the convergence analysis of the polytopal nodal discretisation of contact- mechanics (with Tresca friction) recently introduced in [ Haidar & al, 2024: A bubble vem–fully discrete polytopal scheme for mixed-dimensional poromechanics with frictional contact at matrix-fracture interfaces] in the framework of poro-elastic models in fractured porous media. The scheme is based on a mixed formulation, using face- wise constant approximations of the Lagrange multipliers along the fracture network and a fully discrete first order nodal approximation of the displacement field. The displacement field is enriched with additional bubble degrees of freedom along the fractures to ensure the inf–sup stability with the Lagrange multiplier space. It is presented in a fully discrete formulation, which makes its study more straightforward, but also has a Virtual Element interpretation. The analysis establishes an abstract error estimate accounting for the fully discrete frame- work and the non-conformity of the discretisation. A first order error estimate is deduced for sufficiently smooth solutions both for the gradient of the displacement field and the Lagrange multiplier. A key difficulty of the numerical analysis is the proof of a discrete inf-sup condition, which is based on a non-standard H −1/2 -norm (to deal with fracture networks) and involves the jump of the displacements, not their traces. The analysis also requires the proof of a discrete Korn inequality for the discrete displacement field which takes into account fracture networks. Numerical experiments based on analytical solutions confirm our theoretical findings.

Mots clés: Contact-mechanics fracture networks polytopal method fully discrete approach virtual element method bubble stabilisation error estimates discrete inf-sup condition discrete Korn inequality; Contact-mechanics; fracture networks; polytopal method; fully discrete approach; virtual element method; bubble stabilisation; error estimates; discrete inf-sup condition; discrete Korn inequality

Domaines: Mathématiques [math]

Jérôme Droniou, Ali Haidar, Roland Masson. Analysis of a VEM-fully discrete polytopal scheme with bubble stabilisation for contact mechanics with Tresca friction. 2024. ⟨hal-04542460⟩