Dissertation

Multigrid-based preconditioning for saddle-point problems

Benson, Thomas R

2015

Overview

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, known 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-3 × 3 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 shown.

• Publisher: ProQuest Dissertations & Theses
• Subject: Applied Mathematics
• ISBN: 9781339140155, 1339140152