Premium
Preconditioning a mass‐conserving discontinuous Galerkin discretization of the Stokes equations
Author(s) -
Adler James H.,
Benson Thomas R.,
MacLachlan Scott P.
Publication year - 2017
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2047
Subject(s) - preconditioner , multigrid method , discretization , mathematics , saddle point , discontinuous galerkin method , finite element method , navier–stokes equations , linear system , galerkin method , saddle , relaxation (psychology) , block (permutation group theory) , mathematical analysis , mathematical optimization , compressibility , partial differential equation , geometry , physics , psychology , social psychology , thermodynamics
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 ( P 0 ) 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.