Block Preconditioners for the Marker-and-Cell Discretization of the Stokes-Darcy Equations

TL;DR

This paper develops and analyzes block preconditioners for efficiently solving large saddle-point systems from discretized Stokes-Darcy equations, emphasizing the importance of interface conditions for scalability.

Contribution

It introduces practical preconditioners based on Schur complement approximations that improve convergence and robustness for Stokes-Darcy systems discretized by MAC.

Findings

Preconditioners show good convergence in GMRES.

Robustness against physical parameter variations.

Including interface conditions enhances scalability.

Abstract

We consider the problem of iteratively solving large and sparse double saddle-point systems arising from the stationary Stokes-Darcy equations in two dimensions, discretized by the Marker-and-Cell (MAC) finite difference method. We analyze the eigenvalue distribution of a few ideal block preconditioners. We then derive practical preconditioners that are based on approximations of Schur complements that arise in a block decomposition of the double saddle-point matrix. We show that including the interface conditions in the preconditioners is key in the pursuit of scalability. Numerical results show good convergence behavior of our preconditioned GMRES solver and demonstrate robustness of the proposed preconditioner with respect to the physical parameters of the problem.