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

Domain decomposition preconditioning techniques are developed in the contex t of hp finite element approximation of the Stokes problem. Two basic types of preconditioner are considered: a block diagonal scheme based on decoupl ing the velocity and pressure components, and a scheme based on an indefini te system similar to the original Stokes system. For each type of scheme, t heoretical estimates are obtained for the location of the eigenvalues of th e preconditioned operators in terms of the polynomial degree, the mesh size s on the coarse and fine grids, and the inf-sup constant for the method. Th eoretical estimates show that the growth of the bounds is modest as the mes h 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 pr actical computations for several representative problems. (C) 1999 Elsevier Science S.A. All rights reserved.