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 dimensional porous matrix is coupled to the flow equation on a dimensional fracture surface. The fracture geometry is treated as a possibly non-planar interface, and the fracture is assumed to remain open (, 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 (, 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 模型多速率固定应力分割迭代方案的收敛


本文考虑了裂隙多孔弹性介质中耦合混合维度流和力学问题的收敛分析。在这个混合维度类型系统中，维度多孔基体上的流动方程与维度断裂表面上的流动方程耦合。裂缝几何形状被视为可能的非平面界面，并且假设裂缝保持开放（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力学时间步长内采取多个精细时间步长。在该耦合系统中，多孔基体假设为线性准静态 Biot 模型，断裂假设为润滑型系统（Girault et al., 2015）。更具体地，制定了多速率方案的两种变体（算法）并建立了它们的收敛分析。收敛性分析基于证明 Banach 定点收缩论证，该论证建立了几何收敛性，从而确定了两种算法所获得的解的唯一性。初步的数值结果验证了理论研究结果，对理论研究进行了补充。