带有气泡稳定的接触力学（含Tresca摩擦）VEM-全离散多面体方案分析
Analysis of a VEM-fully discrete polytopal scheme with bubble stabilisation for contact mechanics with Tresca friction
AI总结：
本文对裂隙多孔介质孔隙弹性模型中含Tresca摩擦的接触力学多面体节点离散化进行收敛性分析，提出基于混合公式和气泡自由度稳定的全离散方案，并建立了一阶误差估计与离散inf-sup条件证明。
摘要：
本文在裂隙多孔介质孔隙弹性模型框架下，对文献[18]中最近提出的接触力学（含Tresca摩擦）多面体节点离散化进行了收敛性分析。该方案基于混合公式，沿裂隙网络采用面常数的Lagrange乘子近似，以及位移场的一阶全离散节点近似。位移场沿裂隙被附加的气泡自由度丰富，以确保与Lagrange乘子空间的inf-sup稳定性。它以全离散形式呈现，使其研究更直接，但也具有Virtual Element解释。分析建立了一个抽象误差估计，考虑了全离散框架和离散化的非协调性。对于足够平滑的解，在位移场梯度和Lagrange乘子方面推导出了一阶误差估计。数值分析的一个关键难点是离散inf-sup条件的证明，它基于非标准的$H^{-1/2}$-范数（处理裂隙网络）并涉及位移的跳跃，而非其迹。分析还需要证明考虑裂隙网络的离散位移场的离散Korn不等式。基于解析解的数值实验证实了我们的理论发现。
英文摘要：
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
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