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

翻译：本文针对文献[18]最近在裂隙多孔介质孔隙弹性模型框架下提出的接触力学（含Tresca摩擦）多面体节点离散格式进行收敛性分析。该格式基于混合公式，采用沿裂隙网络的面向常数拉格朗日乘子近似和位移场的全离散一阶节点近似。位移场沿裂隙方向通过添加额外气泡自由度进行丰富，以确保与拉格朗日乘子空间满足inf-sup稳定性条件。该格式以全离散形式呈现，既简化了研究过程，又具有虚拟单元法解释。分析建立了考虑全离散框架和离散非协调性的抽象误差估计。对于充分光滑的解，位移场梯度与拉格朗日乘子均推导出一阶误差估计。数值分析的关键难点在于离散inf-sup条件的证明，该证明基于非标准$H^{-1/2}$范数（用于处理裂隙网络）并涉及位移跳跃量而非迹。分析还要求证明考虑裂隙网络的离散位移场满足离散Korn不等式。基于解析解的数值实验验证了我们的理论结论。