Analysis of a VEM-fully discrete polytopal scheme with bubble stabilisation for contact mechanics with Tresca friction
<|reference_start|>Analysis of a VEM-fully discrete polytopal scheme with bubble stabilisation for contact mechanics with Tresca friction: This work performs the convergence analysis of the polytopal nodal discretisation of contact-mechanics (with Tresca friction) recently introduced in [18] 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 framework 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<|reference_end|>
@article{droniou2024analysis,
    title={Analysis of a VEM-fully discrete polytopal scheme with bubble
  stabilisation for contact mechanics with Tresca friction},
    author={J'er^ome Droniou, Ali Haidar, Roland Masson},
    journal={arXiv preprint arXiv:2404.03045},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.03045},
    primaryClass={math.NA cs.NA}
}