Title: Analysis of a VEM-fully discrete polytopal scheme with bubble stabilisation for contact mechanics with Tresca friction Show Chinese title

Title: 接触力学中具有Tresca摩擦的VEM全离散多面体格式的分析，带有泡状稳定化

Abstract: 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 Show Chinese abstract

Abstract: 这项工作对最近在[18]中引入的接触力学（Tresca摩擦）的多面体节点离散化进行了收敛性分析，在裂缝多孔介质的渗流弹性模型框架内。 该方案基于一种混合公式，使用沿裂缝网络的面上常数近似来表示拉格朗日乘子，并采用位移场的完全离散的一阶节点近似。 位移场在裂缝处增加了额外的泡状自由度，以确保与拉格朗日乘子空间的inf-sup稳定性。 它以完全离散的形式呈现，这使得研究更加直接，同时也具有虚拟元素解释。 分析建立了一个抽象的误差估计，考虑了完全离散框架和离散化的非一致性。 对于足够光滑的解，得出了一阶误差估计，适用于位移场的梯度和拉格朗日乘子。 数值分析的一个关键难点是证明一个离散的inf-sup条件，该条件基于一种非标准的$H^{-1/2}$-范数（用于处理裂缝网络），并涉及位移的跳跃，而不是它们的迹。 分析还要求证明一个离散的Korn不等式，适用于考虑裂缝网络的离散位移场。 基于解析解的数值实验验证了我们的理论结果。