We consider preconditioning methods for convection-dominated fluid flow problems based on a nonoverlapping Schur complement domain decomposition procedure for arbitrary triangulated domains. The triangulation is first partitioned into a number of subdomains and interfaces which induce a natural $2 \times 2$ partitioning of the p.d.e. discretization matrix. We view the Schur complement induced by this partitioning as an algebraically derived coarse space approximation. This avoids the known difficulties associated with the direct formation of an effective coarse discretization for advection dominated equations. By considering various approximations of the block factorization of the $2 \times 2$ system, we have developed a family of robust preconditioning techniques.
These approximations are introduced to improve both the sequential and parallel efficiency of the method without significantly degrading the quality of the preconditioner. The specific approximations that we have used include ILU-preconditioned GMRES subdomain solves, localized approximation of the interface Schur complement, and limited level-fill ILU interface backsolves. A computer code based on these ideas has been developed and tested on the IBM SP2 using MPI message passing protocol. A number of 2-D CFD calculations will be presented for both scalar advection-diffusion equations and the Euler equations. These results show very good scalability of the preconditioner as the number of processors is increased while the number of degrees of freedom per processor is fixed.
This is joint work with Tim Barth and Wei-Pei Tang.