Commun. Comput. Phys., 16 (2014), pp. 1263-1297.


Efficient Variable-Coefficient Finite-Volume Stokes Solvers

Mingchao Cai 1, Andy Nonaka 2, John B. Bell 2, Boyce E. Griffith 3, Aleksandar Donev 1*

1 Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA.
2 Center for Computational Sciences and Engineering, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA.
3 Leon H. Charney Division of Cardiology, Department of Medicine, New York University School of Medicine, NY, USA; Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA; Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA.

Received 7 January 2014; Accepted (in revised version) 17 June 2014
Available online 29 August 2014
doi:10.4208/cicp.070114.170614a

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. 7565-7595], 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.

AMS subject classifications: 65F08, 65F10

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Key words: Stokes flow, variable density, variable viscosity, saddle point problems, projection method, preconditioning, GMRES.

*Corresponding author.
Email: cmchao2005@gmail.com (M. Cai), ajnonaka@lbl.gov (A. Nonaka), jbbell@lbl.gov (J. B. Bell), boyceg@email.unc.edu (B. E. Griffith), donev@courant.nyu.edu (A. Donev)
 

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