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]在裂隙多孔介质中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。该格式采用混合形式，沿裂隙网络使用逐面常数近似拉格朗日乘子，并对位移场采用全离散一阶节点逼近。位移场沿裂隙方向通过附加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，使得研究更为直接，同时具备虚拟单元法解释。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。对于足够光滑的解，推导了位移场梯度和拉格朗日乘子的一阶误差估计。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），涉及位移的跳跃量而非迹线。分析还需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果。