Preconditioning a mass‐conserving discontinuous Galerkin discretization of the Stokes equations

• Published in: Numerical linear algebra with applications 2017-05, Vol.24 (3), p.np-n/a
• Main Authors: Adler, James H., Benson, Thomas R., MacLachlan, Scott P.
• Format: Article
• Language: English
• Subjects: block‐factorization preconditioners; Computational fluid dynamics; Discontinuity; discontinuous Galerkin; Discretization; Fluid flow; Galerkin methods; Mathematical analysis; Mathematical models; monolithic multigrid; Navier-Stokes equations; Stokes equations; Stokes law (fluid mechanics)
• ISSN: 1070-5325; 1099-1506
• DOI: 10.1002/nla.2047

Summary The incompressible. Stokes equations are a widely used model of viscous or tightly confined flow in which convection effects are negligible. In order to strongly enforce the conservation of mass at the element scale, special discretization techniques must be employed. In this paper, we consider a discontinuous Galerkin approximation in which the velocity field is H(div,Ω)‐conforming and divergence‐free, based on the Brezzi, Douglas, and Marini finite‐element space, with complementary space (P0) for the pressure. Because of the saddle‐point structure and the nature of the resulting variational formulation, the linear systems can be difficult to solve. Therefore, specialized preconditioning strategies are required in order to efficiently solve these systems. We compare the effectiveness of two families of preconditioners for saddle‐point systems when applied to the resulting matrix problem. Specifically, we consider block‐factorization techniques, in which the velocity block is preconditioned using geometric multigrid, as well as fully coupled monolithic multigrid methods. We present parameter study data and a serial timing comparison, and we show that a monolithic multigrid preconditioner using Braess–Sarazin style relaxation provides the fastest time to solution for the test problem considered. Copyright © 2016 John Wiley & Sons, Ltd.