| 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. A convergence analysis is then given for dual-primal FETI algorithms both in two and in three dimensions.; Extension to solving linearized non-symmetric Navier-Stokes equation is also discussed. The resulting linear system is no longer symmetric and a GMRES method is used to solve this preconditioned linear system. Eigenvalue estimate shows that, for small Reynolds number, the non-symmetric 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.; The algorithms are further extended to nonlinear 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 show that the convergence of Picard iteration depends on the Reynolds number, but is independent of both the number of subdomains and the subdomain problem size. |
