Discontinuous Galerkin methods with solenoidal elements for the Stokes problem

Discontinuous Galerkin methods with solenoidal elements for the Stokes problem

Heemskerk, T.K.
Van Brummelen, E.H.
Van der Zee, K.G.

Aerospace Engineering

Aerospace Materials & Manufacturing

2006-11-01

The essential difficulty in the numerical solution of the incompressible Navier-Stokes (NS) equations is the coupling between the pressure and the velocity. The coupling enforces a constraint on the relation between the pressure and the velocity space. This constraint can be studied using a simplified form of the NS equations, viz. the Stokes equations. The present work examines the applicability of discontinuous Galerkin (DG) methods to the Stokes equations. Moreover, the application of DG-methods with locally solenoidal velocity fields is investigated. One of the main advantages of using DG-methods is their suitability to hp-adaptivity, as they can easily handle complex geometries and approximations that have polynomials of different degrees in different elements. We have restricted ourselves to the two most prominent DG methods for the Laplace operator, viz. the asymmetric Baumann-Oden (BO) method and the Symmetric Interior Penalty Galerkin (SIPG) method. This work presents stability analysis for various elements with diverse finite-element spaces, together with analytically and numerically determined error convergence rates. Also a measure for the computational effort is given for each element. Emphasis is put on varying combinations of polynomial orders for the velocity and pressure spaces, showing the possibilities for adaptivity. Using the results of these tests, we see that applicability of DG-methods for the Stokes problem is very promising, especially with respect to stability, implementation of non-homogeneous boundary conditions and adaptivity. Taking into account the error convergence, computational cost and the adaptivity properties, the solenoidal DG-element with discontinuous pressure on the skeleton is the most promising element. For this element the SIPG-method is the best method, because it is more stable and convergence is somewhat better than the BO-method, unless a penalty parameter is undesirable. In that case one should opt for the BO-method.

Stokes problem
finite element methods
solenoidal velocity field

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Delft Aerospace Computational Science

1574-6992

Report DACS-06-006

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(c) 2006 The Author(s)