Schwarz preconditioners for the spectral element discretization of the steady Stokes and Navier-Stokes equations

The Q(N_)Q(N-2) spectral element discretization of the Stokes equation give s rise to an ill-conditioned. indefinite, symmetric linear system for the v elocity and pressure degrees of freedom. We propose a domain decomposition method which involves the solution of a low-order global, and several local problems, related to the vertices, edges, and interiors of the subdomains. The original system is reduced to a symmetric equation for the velocity. w hich can be solved with the conjugate gradient method. We prove that the co ndition number of the iteration operator is bounded from above by C(1 + log (N))(3/)beta (n), where C is a positive constant independent of the degree N and the number of subdomains, and beta (N) is the inf-sup condition of th e pair Q(N-)Q(n-2).We also consider the stationary Navier-Stokes equations; , in each Newton step, a non-symmetric indefinite problem is solved using a Schwarz preconditioner. By using an especially designed low-order global s pace, and the solution of local problems analogous to those decribed above for the Stokes equation, we are able to present a complete theory for the m ethod. We prove that the number of iterations of the GMRES method, at each Newton step, is bounded from above by C(1 + log(N))(3)/beta (N). The consta nt C does not depend on the number of subdomains or N. and it does not dete riorate as the Newton iteration proceeds.