Domain decomposition preconditioners for p and hp finite element approximation of Stokes equations

Abstract

Domain decomposition preconditioning techniques are developed in the context of hp finite element approximation of the Stokes problem. Two basic types of preconditioner are considered: a block diagonal scheme based on decoupling the velocity and pressure components, and a scheme based on an indefinite system similar to the original Stokes system. For each type of scheme, theoretical estimates are obtained for the location of the eigenvalues of the preconditioned operators in terms of the polynomial degree, the mesh sizes on the coarse and fine grids, and the inf—sup constant for the method. Theoretical estimates show that the growth of the bounds is modest as the mesh is refined and the polynomial order is increased. The preconditioners are shown to be applicable to various iterative schemes for the Stokes systems. The theoretical bounds are compared with actual quantities obtained in practical computations for several representative problems.