by Schieweck, F..

**Series:** 2002-34, Preprints

- MSC:
- 65N30 Finite elements, ~Rayleigh-Ritz and Galerkin methods, finite methods
- 65N22 Solution of discretized equations
- 76D07 Stokes and related (Oseen, etc.) flows
- 76D05 ~Navier-Stokes equations

**Abstract:**

We consider an $hp$-like finite element method on a 2D quadrilateral mesh

for solving the Stokes problem with continuous $Q_r$-elements for the

velocity and discontinuous $P_{r-1}$-elements for the pressure where the

order $r$ can vary from element to element.

We construct a local

and practically useful basis for the subspace of the discretely

divergence free velocity functions.

Thus, the computation of the velocity can be reduced to the solution

of a symmetric positive definite problem in this subspace.

Our constructed basis is suitable to create a multigrid method for

solving the subspace problem efficiently.

The pressure can be

obtained by a simple post-processing procedure

which requires only the solution of local problems.

The ideas for the construction of a discretely divergence free

basis can be extended easily to the general case of an adaptive

mixed mesh with triangular and quadrilateral elements and hanging

nodes.

The approach can be applied also to the incompressible Navier-Stokes

equations.

**Keywords:**

higher order finite elements, discretely divergence free basis, incompressible flow, Stokes equations, Navier-Stokes equations