Efficient Variable-Coefficient Finite-Volume Stokes Solvers (Communications in Computational Physics, 2014)
We investigate several robust preconditioners for solving the saddle-pointlinear systems that arise from spatial discretization of unsteady and steady variable-coefficient Stokes equations on a uniform staggered grid. Building on the success ofusing 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 asestablished techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtzand Poisson solvers for the velocity and pressure subproblems. We demonstrate thatonly a single cycle of a standard geometric multigrid algorithm serves as an effectiveinexact solver for each of these subproblems. Contrary to traditional wisdom, we findthat the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes systemcomparable 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 viscositycontrasts. Two of the five preconditioners considered here are found to be robust toGMRES restarts and to increasing problem size, making them suitable for large-scaleproblems. Our work opens many possibilities for constructing novel unsplit temporalintegrators for finite-volume spatial discretizations of the equations of low Mach andincompressible flow dynamics.