This paper considers the convergence analysis of a coupled mixed dimensional flow and mechanics problem in a fractured poro-elastic medium. In this mixed dimensional type system, the flow equation on a d dimensional porous matrix is coupled to the flow equation on a d−1 dimensional fracture surface. The fracture geometry is treated as a possibly non-planar interface, and the fracture is assumed to remain open (i.e., ignoring contact mechanics boundary conditions). The main contribution is developing a multirate scheme in which flow takes multiple fine time steps within one coarse mechanics time step using the standard fixed stress splitting approach. In this coupled system, the linear quasi-static Biot model is assumed for the porous matrix, and the lubrication-type system (Girault et al., 2015) is assumed for the fracture. More specifically, two variations (i.e., algorithms) of the multirate scheme are formulated and their convergence analyses are established. The convergence analysis is based on proving a Banach fixed-point contraction argument which establishes the geometric convergence, and hence, the uniqueness of the obtained solution for both algorithms. The theoretical investigations are supplemented by preliminary numerical results validating the theoretical findings.

中文翻译：

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断裂 Biot 模型的多速率固定应力分裂迭代方案的收敛


本文考虑了裂隙多孔弹性介质中混合维度流动和力学耦合问题的收敛分析。在这个混合维度类型系统中，d 维多孔基体上的流动方程与 d-1 维裂隙表面上的流动方程耦合。裂缝几何被视为可能的非平面界面，并假设裂缝保持开放（即忽略接触力学边界条件）。主要贡献是开发了一种多速率方案，其中流动使用标准固定应力分裂方法在一个粗略力学时间步长内采用多个精细时间步长。在这个耦合系统中，多孔基体假设线性准静态 Biot 模型，裂缝假设润滑型系统（Girault et al.， 2015）。更具体地说，制定了多速率方案的两种变体 （即算法） 并建立了它们的收敛分析。收敛分析基于证明 Banach 定点收缩论点，该论点建立了几何收敛性，因此，两种算法所获得的解决方案的唯一性。理论研究得到了验证理论发现的初步数值结果的补充。