Title: Dual-Primal FETI Methods for Stationary
Stokes and Navier-Stokes Equations

(NYU-CS-TR830)

Author: Jing Li, Courant Institute of Mathematical Sciences

Abstract:
Finite element tearing and interconnecting (FETI) type domain decomposition 
methods are first extended to solving incompressible Stokes equations.
One-level, two-level, and dual-primal FETI algorithms are proposed.
Numerical experiments show that these FETI type algorithms are scalable, i.e., 
the number of iterations is independent of the number of subregions into which 
the given domain is subdivided. A convergence analysis is then given for 
dual-primal FETI algorithms both in two and three dimensions. 

Extension to solving linearized nonsymmetric stationary Navier-Stokes equations is 
also discussed. The resulting linear system is no longer symmetric and a GMRES 
method is used to solve the preconditioned linear system. Eigenvalue estimates show 
that, for small Reynolds number, the nonsymmetric preconditioned linear system 
is a small perturbation of that in the symmetric case. Numerical experiments also 
show that, for small Reynolds number, the convergence of GMRES method is similar 
to the convergence of solving symmetric Stokes equations with the conjugate 
gradient method. The convergence of GMRES method depends on the Reynolds number; 
the larger the Reynolds number, the slower the convergence. 

Dual-primal FETI algorithms are further extended to nonlinear stationary Navier-Stokes 
equations, which are solved by using a Picard iteration. 
In each iteration step, a linearized Navier-Stokes equation is solved by using a 
dual-primal FETI algorithm. Numerical experiments indicate that convergence of 
the Picard iteration depends on the Reynolds number, but is independent of both the 
number of subdomains and the subdomain problem size.