PhD, September 2004
Advisor: Xiao-Chuan Cai
thesis.pdf
Domain decomposition methods are widely used and very powerful for solving large sparse linear and nonlinear systems of equations arising from partial differential equations (PDEs). Among different families of domain decomposition methods, we focus primarily on the class of Schwarz type methods. This dissertation proposes and tests some general techniques of linear and nonlinear preconditioning based on a Schwarz framework for two such challenging problems, namely the Stokes problem and incompressible Navier-Stokes equations in computational fluid dynamics.
The two-level preconditioners comprise two parts: local additive Schwarz preconditioners, which are constructed by using the solution of discrete PDEs defined on the overlapping subdomain with some proper boundary conditions, and a global coarse preconditioner, which is defined by the solution on coarse meshes of either original discrete PDEs for the Stokes problem or the linear approximation of PDEs for incompressible Navier-Stokes equations. The two-level preconditioners are applied in conjunction with some linear or nonlinear iterative methods, such as Krylov subspace methods or Newton methods.
Numerical results obtained on parallel computers show that (1) for the Stokes problem, the performance of the two-level method with a multiplicative coarse preconditioner is superior to the other two variants of additive Schwarz preconditioners; (2) for incompressible Navier-Stokes equations, the local nonlinear preconditioners make the Newton method more robust in the sense that the method converges within few iterations for a wide range of Reynolds numbers and mesh sizes, and the linear coarse preconditioner makes the method more scalable in the sense that the number of linear iterations depends only slightly on the number of parallel processors.