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：本文对近期文献[18]在裂缝多孔介质孔隙弹性模型框架中提出的接触力学（含Tresca摩擦）多面体节点离散格式进行了收敛性分析。该格式基于混合形式，采用沿裂缝网络的逐面常数近似拉格朗日乘子，以及位移场的全离散一阶节点近似。位移场沿裂缝位置引入额外气泡自由度以增强其与拉格朗日乘子空间的inf-sup稳定性。该格式采用全离散形式表述，简化了研究过程，同时具备虚拟元解释。分析建立了适用于全离散框架及离散非一致性的抽象误差估计。对于梯度足够光滑的位移场与拉格朗日乘子解，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，该条件基于非标准$H^{-1/2}$范数（适用于裂缝网络），涉及位移跳跃量而非迹。分析还需证明考虑裂缝网络的离散Korn不等式。基于解析解的数值实验验证了理论结果。