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:我们考虑Biot模型,其中含有块状先决条件和通用的半数值问题,以适应和稳妥性参数。使用不连续的 Galerkin 离散法,在 $P_1 元元素中采用离位和达西流通分解法,作为小块连续,在 $P_0 元元素中采用小块常态,在稳定期中使用 $P_0 元元素。平行算法旨在解决由此产生的线性系统。具体地说,GMRES 方法是作为外部循环算法和块状角前置法而使用的,目的是加快趋同速度。在前提中,使用不完整的Choolesky因子化法或双级叠加法来进一步接近离子操作者。在这种方法中,粗网是由重叠(GenEO)中通用的单数值问题构建的。广泛的数字实验显示了由此产生的平行算法的可伸缩性和准性。