A fast domain decomposition solver for the discretized Stokes equations by a stabilized finite element method

Dr. Atsushi Suzuki ( Czech Technical University in Prague / Kyushu University )

An iterative substructuring method with balancing Neumann-Neumann preconditioner is known as an efficient parallel algorithm for the elasticity equations. This method was extended to the Stokes equations by Pavarino and Widlund [2002]. In their extension, Q2/P0-discontinuous elements are used for velocity/pressure and a Schur complement system within "benign space", where incompressibility satisfied, is solved by CG method.

For the construction of the coarse space for the balancing preconditioner, some supplementary solvability conditions are considered. In our algorithm for 3-D computation, P1/P1 elements for velocity/pressure with pressure stabilization are used to save computational cost in the stiffness matrix. We introduce a simple coarse space similar to the one of elasticity equations. Owing to the stability term, solvabilities of local Dirichlet problem for a Schur complement system, of Neumann problem for the preconditioner, and of the coarse space problem are ensured. In our implementation, local Dirichlet and Neumann problems are solved by a 4x4-block modified Cholesky factorization procedure with an envelope method, which leads to fast computation with small memory requirement. Numerical result on parallel efficiency with a shared memory computer will be shown. Direct use of the Stokes solver in an application of Earth's mantle convection problem will be also shown.