Print Email Facebook Twitter Discontinuous Galerkin methods with solenoidal elements for the Stokes problem Title Discontinuous Galerkin methods with solenoidal elements for the Stokes problem Author Heemskerk, T.K. Van Brummelen, E.H. Van der Zee, K.G. Faculty Aerospace Engineering Department Aerospace Materials & Manufacturing Date 2006-11-01 Abstract 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. Subject Stokes problemfinite element methodssolenoidal velocity field To reference this document use: http://resolver.tudelft.nl/uuid:9f5d9377-c732-45ad-8d41-d5fda2e05ec9 Publisher Delft Aerospace Computational Science ISSN 1574-6992 Source Report DACS-06-006 Part of collection Institutional Repository Document type report Rights (c) 2006 The Author(s) Files PDF Heemskerk_2006.pdf 946.13 KB Close viewer /islandora/object/uuid:9f5d9377-c732-45ad-8d41-d5fda2e05ec9/datastream/OBJ/view