Efficient parameter-robust preconditioners for linear poroelasticity and elasticity in the primal formulation

TL;DR

This paper introduces new nonsingular preconditioners for linear poroelasticity and elasticity problems that are robust against parameter variations and effective in accelerating iterative solutions, especially in locking cases.

Contribution

The authors develop parameter-robust, nonsingular preconditioners that do not require Schur complement computation, improving the efficiency of solving large-scale saddle-point systems in poroelasticity.

Findings

Eigenvalues cluster around 1 with an outlier of order 1/λ.

Preconditioners are robust to mesh size, time step, and locking parameters.

Numerical results confirm effectiveness and robustness.

Abstract

Poroelasticity problems play an important role in various engineering, geophysical, and biological applications. Their full discretization results in a large-scale saddle-point system at each time step that is becoming singular for locking cases and needs effective preconditioners for its fast iterative solution. Instead of constructing spectrally equivalent ones, we develop nonsingular preconditioners so that the eigenvalues of the preconditioned system consist of a cluster around $1$ and an outlier in the order of $1/\lambda$, where $\lambda$ is a Lam\'{e} constant that is large for locking cases. It is known that the convergence factor of GMRES is bounded by the radius of the cluster for this type of systems. Both two- and three-field block triangular Schur complement preconditioners are studied. Upper bounds of the radius of the eigenvalue cluster for those systems are obtained and shown to be related to the inf-sup condition but independent of mesh size, time step, and locking parameters, which reflects the robustness of the preconditioners with respect to parameter variations. Moreover, the developed preconditioners do not need to compute the Schur complement and neither require exact inversion of diagonal blocks except the leading one. A locking-free weak Galerkin finite element method and the implicit Euler scheme are used for the discretization of the governing equation. Both two- and three-dimensional numerical results are presented to confirm the effectiveness and parameter-robustness of the developed preconditioners.