On the Interface and Two-Level Preconditioners in Newton-Schwarz Method

This paper is concerned with parallel computation in solving the convection-diffusion equation and the incompressible Navier-Stokes equation via Newton-Schwarz method, a nonlinear domain decomposition (DD) method. Various preconditioners are investigated here. An interface problem is tackled as a preconditioner for nonlinear block Jacobi DD approach, with an optional fine level interface problem solved as further preconditioner. Also a (global) coarse level preconditioner is considered. Examined also is the relaxation type preconditioner. Such preconditioned nonlinear DD methods exhibit impressive improvement over the basic non-preconditioned parallel NewtonJacobi method. A general review on Newton-Schwarz method is [GEMT98]. Our setup has the advantages of both the overlapped and the nonoverlap DD approach. The subdomain variables form a (nonoverlap) partition of the whole global system of equations. Solving the interface problem is regarded as a preconditioner to all subproblems. We note that the interface variables were excluded from subproblems in [LC98]. We describe in later sections the problem and solution procedure, the test cases and results, and a brief conclusion.