Abstract
We study the performance of some preconditioning techniques for a class of block three-by-three linear systems of equations arising from finite element discretizations of the coupled Stokes–Darcy flow problem. In particular, we investigate preconditioning techniques including block preconditioners, constraint preconditioners, and augmented Lagrangian-based ones. Spectral and field-of-value analyses are established for the exact versions of these preconditioners. The result of numerical experiments are reported to illustrate the performance of inexact variants of the various preconditioners used with flexible GMRES in the solution of a 3D test problem with large jumps in the permeability.
Highlights
The coupled Stokes–Darcy model describes the interaction between free flow and porous media flow
First we present the results of experiments in which, inside FGMRES, the symmetric positive definite (SPD) subsystems were solved inexactly by the preconditioned conjugate gradient (PCG) method using loose tolerances
Under “Iterpcgi ” (“Itercgi ”) we further report the total number of inner PCG iterations performed for solving the linear systems corresponding to block (i, i) of the preconditioner, where i = 1, 2
Summary
The coupled Stokes–Darcy model describes the interaction between free flow and porous media flow. P2 represents the Darcy pressure in 2, and the symmetric positive definite (SPD) matrix K represents the hydraulic conductivity in the porous medium. The Field-of-Values (FOV) equivalence of constraint preconditioners with A was proved in [13] It is well-known that if a preconditioner is norm equivalent to the coefficient matrix of a linear system of equations, the spectra of the preconditioned system remain uniformly bounded and bounded away from zero as the mesh size h → 0, see [23] for more details. The linear system of equations Ax = bis equivalent to Au = b This approach is motivated by the success of the use of grad-div stabilization and augmented Lagrangian techniques for solving saddle point problems.
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