A Dual-Primal FETI Method for Three-Dimensional Incompressible Stokes Equations

Jing Li

Courant Institute of Mathematical Sciences
New York University
251 Mercer Street
New York, NY 10012, USA


Abstract

In this paper, a dual-primal FETI method is developed for solving incompressible Stokes equations approximated by mixed finite elements with discontinuous pressures in three dimensions. The domain of the problem is decomposed into non-overlapping subdomains, and the continuity of the velocity across the subdomain interface is enforced by introducing Lagrange multipliers. By a Schur complement procedure, solving the indefinite Stokes problem is reduced to solving a symmetric positive definite problem for the dual variables, i.e., the Lagrange multipliers. This dual problem is solved by a Krylov space method with a Dirichlet preconditioner. At each step of the iteration, both subdomain problems and a coarse problem on a coarse subdomain mesh are solved by a direct method. It is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from aboveby the product of the inverse of the inf-sup constant of the discrete problem and the square of the logarithm of the number of unknownsin the individual subdomain problems. Illustrative numerical results are presented by solving a three-dimensional lid driven cavity problem.

Key words. domain decomposition, Stokes, FETI, dual-primal methods

AMS subject classifications. 65N30, 65N55, 76D07