@Article{CiCP-16-1263,
author = {Mingchao Cai, Andy Nonaka, John B. Bell, Boyce E. Griffith and Aleksandar Donev},
title = {Efficient Variable-Coefficient Finite-Volume Stokes Solvers},
journal = {Communications in Computational Physics},
year = {2014},
volume = {16},
number = {5},
pages = {1263--1297},
abstract = {<p style="text-align: justify;">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 [<em>B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 7565–7595</em>], 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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.070114.170614a},
url = {http://global-sci.org/intro/article_detail/cicp/7080.html}
}