Biot model with generalized eigenvalue problems for scalability and robustness to parameters

TL;DR

This paper develops scalable, robust parallel algorithms for the Biot model using block preconditioners, generalized eigenvalue problems, and discontinuous Galerkin discretization, demonstrating effectiveness through extensive numerical experiments.

Contribution

It introduces a novel combination of block preconditioners, generalized eigenvalue problems, and discontinuous Galerkin methods for improved scalability and robustness in Biot model simulations.

Findings

Algorithms are scalable and robust to parameter variations.

Parallel GMRES with block-triangular preconditioners accelerates convergence.

Numerical experiments confirm effectiveness and robustness.

Abstract

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.