We consider Biot model with block preconditioners and generalized eigenvalue problems for scalability and robustness to parameters. A discontinuous Galerkin discretization is employed with the displacement and Darcy flow flux discretized as piecewise continuous in $P_1$ elements, and the pore pressure as piecewise constant in the $P_0$ element with a stabilizing term. Parallel algorithms are designed to solve the resulting linear system. Specifically, the GMRES method is employed as the outer iteration algorithm and block-triangular preconditioners are designed to accelerate the convergence. In the preconditioners, the elliptic operators are further approximated by using incomplete Cholesky factorization or two-level additive overlapping Schwartz method where coarse grids are constructed by generalized eigenvalue problems in the overlaps (GenEO). Extensive numerical experiments show a scalability and parametric robustness of the resulting parallel algorithms.

翻译：本文考虑基于块预条件子和广义特征值问题的Biot模型，以实现可扩展性和参数鲁棒性。采用间断伽辽金离散方法，将位移和达西流通量在P1单元中进行分段连续离散，孔隙压力在P0单元中以带稳定项的分段常数形式离散。设计了并行算法求解所得线性系统：外层迭代采用GMRES方法，并设计块三角预条件子加速收敛。预条件子中，椭圆算子分别通过不完全乔莱斯基分解或基于重叠区域广义特征值构造粗网格的两层加性施瓦茨方法（GenEO）进行近似。大量数值实验表明，所提并行算法具有良好的可扩展性和参数鲁棒性。