Comparisons of Some Iterative Algorithms for Biot Equations

TL;DR

This paper compares iterative algorithms, including GMRES, Uzawa, and Anderson acceleration, for solving stabilized finite element discretizations of the Biot equations, focusing on their efficiency and effectiveness.

Contribution

The paper introduces and compares multiple iterative methods with specialized preconditioners for solving Biot equations, highlighting their performance differences.

Findings

GMRES outperforms other methods in convergence speed.

Preconditioners based on Schur complement approximation improve solver efficiency.

Anderson acceleration enhances the convergence of Uzawa method.

Abstract

In this paper, we aim at solving the Biot model under stabilized finite element discretizations. To solve the resulting generalized saddle point linear systems, some iterative methods are proposed and compared. In the first method, we apply the GMRES algorithm as the outer iteration. In the second method, the Uzawa method with variable relaxation parameters is employed as the outer iteration method. In the third approach, Uzawa method is treated as a fixed-point iteration, the outer solver is the so-called Anderson acceleration. In all these methods, the inner solvers are preconditioners for the generalized saddle point problem. In the preconditioners, the Schur complement approximation is derived by using Fourier analysis approach. These preconditioners are implemented exactly or inexactly. Extensive experiments are given to justify the performance of the proposed preconditioners and to compare all the algorithms.