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Construction of Higher Order Discretely Divergence Free Finite Elements for Incompressible Flow

by Schieweck, F..

Series: 2002-34, Preprints

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

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

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