Schwarz type solvers for <em>hp</em>-FEM discretizations of mixed problems

Abstract

The Stokes problem and linear elasticity problems can be viewed as a mixed variational formulation. These formulations are discretized by means of the <em>hp</em>-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 inf-sup 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 <em>hp</em>-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 pressure is discretized by discontiuous finite elements. We will prove that the system of linear algebraic equations can be solved in almost optimal complexity if the <em>Q_k</em> − <em>P_{k</em>_-1,<em>disc}</em> element is used. This yields to quasioptimal <em>hp</em>-FEM solvers for the Stokes problems and linear elasticity problems. The latter are robust with respect to the contraction ratio <em>ν</em>. The efficiency of the presented solver is shown in several numerical examples.