Multigrid-based preconditioning for saddle-point problems.
Abstract: Saddle-point problems arise in a variety of applications, from
finance to physics, including liquid crystals research, solid mechanics, and fluid
dynamics. These linear systems are often challenging to solve due to their indefiniteness,
and many common preconditioners yield poor performance or fail altogether, unless closely
tailored to the application. Multigrid methods are, however, ... read moreknown to provide efficient,
optimal methods for a variety of problems. Here, we consider multigrid-based
preconditioning techniques for saddle-point problems that arise in fluid dynamics
simulations, primarily focusing on the use of monolithic multigrid methods that treat all
variables in the system at once. The first application we consider is the numerical
solution of the incompressible Stokes equations. This system is used to model low Reynolds
number flows that are very viscous or tightly confined, such as in geophysical or
hemodynamic simulations. We explore a discontinuous Galerkin finite-element discretization
in which the resulting velocity field is exactly divergence-free. Due to the saddle-point
structure of the resulting linear system and the complex nature of the discretization,
specialized preconditioning methods are required. Here, we compare block-factorization
preconditioners, using multigrid as an approximate inverse for the velocity block, with
fully-coupled multigrid preconditioners that utilize extended versions of well-known
relaxation techniques. Parameter studies for each of these preconditioners as well as
serial timing studies are shown. The second application is magnetohydrodynamics (MHD),
which couples the incompressible Navier-Stokes equations with Maxwell's equations and is
used to model the behavior of a charged fluid in the presence of electromagnetic fields.
This is a system of nonlinear partial differential equations; thus, we use a Newton-Krylov
approach and study the use of monolithic multigrid preconditioners for the linear systems
that arise from the linearization and finite-element discretization of the problem. We
first consider a vector-potential formulation of resistive MHD, and extend the well-known
Vanka and Braess-Sarazin relaxation schemes to the case of this block-3x3 saddle-point
problem. After showing parameter and timing studies for this problem, we extend these
approaches to a discretization that uses a second Lagrange multiplier to enforce the
solenoidal constraint. Numerical studies for a variety of test problems are
Thesis (Ph.D.)--Tufts University, 2015.
Submitted to the Dept. of Mathematics.
Advisor: James Adler.
Committee: Bruce Boghosian, Xiaozhe Hu, Raymond Tuminaro, and Scott MacLachlan.
Keyword: Applied mathematics.read less