Schwarz Type Solvers for hp-FEM Discretizations of Mixed Problems

The Stokes problem and linear elasticity problems can be viewed as a mixed variational formulation. These formulations are discretized by means of the hp-version of the finite element method. The system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method. The development an efficient preconditioner requires three ingredients, a preconditioner related to the components of the velocity modes, a preconditioner for the Schur complementrelated to the components of the pressure modes and the discrezation by a stable finite element pair which satisfies the discrete infsup condition. The last condition is also important in order to obtain a stable discretization scheme. The preconditioner for the velocity modes is adapted from fast hp-FEM preconditioners for elliptic problems. Moreover, we will prove that the preconditioner for the Schur complement can be chosen as a diagonal matrix if the pressureis discretized by discontiuous finite elements.We will prove that the system of linear algebraic equationscan be solved in almost optimal complexity if the Qk − Pk−1,disc element is used. This yields to quasioptimal hp-FEM solvers for the Stokes problems and linear elasticity problems. The latter are robust with respect to the contraction ratio ν. The efficiency of the presented solver is shown in several numerical examples.