Triangular preconditioners for double saddle point linear systems arising in the mixed form of poroelasticity equations

TL;DR

This paper introduces inexact block triangular preconditioners for double saddle-point systems from poroelasticity equations, providing spectral analysis and demonstrating improved efficiency over existing methods through numerical experiments.

Contribution

It develops a spectral analysis of new preconditioners for double saddle-point systems in poroelasticity, enhancing iterative solver performance.

Findings

Eigenvalues lie in a circle of radius less than 1

Preconditioners improve GMRES convergence

Numerical results confirm theoretical bounds

Abstract

In this paper, we study a class of inexact block triangular preconditioners for double saddle-point symmetric linear systems arising from the mixed finite element and mixed hybrid finite element discretization of Biot's poroelasticity equations. We develop a spectral analysis of the preconditioned matrix, showing that the complex eigenvalues lie in a circle of center $(1,0)$ and radius smaller than 1. In contrast, the real eigenvalues are described in terms of the roots of a third-degree polynomial with real coefficients. The results of numerical experiments are reported to show the quality of the theoretical bounds and illustrate the efficiency of the proposed preconditioners used with GMRES, especially in comparison with similar block diagonal preconditioning strategies along with the MINRES iteration.