02-34
Construction of Higher Order Discretely Divergence Free Finite Elements for Incompressible Flow
Preprint series: 02-34, Preprints
- MSC:
- 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
- 65N22 Solution of discretized equations, See also {65Fxx, 65Hxx}
- 76D07 Stokes flows
- 76D05 Navier-Stokes equations, See also {35Q30}
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
The author(s) agree, that this abstract may be stored asfull text and distributed as such by abstracting services.