Efficient Variable-Coefficient Finite-Volume Stokes Solvers

TL;DR

This paper develops efficient preconditioners for variable-coefficient Stokes equations, enabling nearly optimal solution times comparable to classical methods, even for large-scale and complex flow problems.

Contribution

It introduces robust preconditioners using Helmholtz and Poisson solvers, demonstrating near-optimal efficiency for solving saddle-point systems in variable-coefficient Stokes flow.

Findings

Single multigrid cycle effectively preconditions subproblems

Stokes system solved nearly as efficiently as separate velocity and pressure problems

Preconditioners are robust to problem size and GMRES restarts

Abstract

We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variable-coefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocity-pressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 75657595], as well as established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts. Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size, making them suitable for large-scale problems. Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.