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Preprint 11-2013

Finite Element Decomposition and Minimal Extension for Flow Equations

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Author(s) : Robert Altmann , Jan Heiland

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 11-2013

MSC 2000

76M10 Finite element methods
65L80 Methods for differential-algebraic equations
65J10 Equations with linear operators

Abstract :
In the simulation of flows, the correct treatment of the pressure variable is the key to stable time-integration schemes. This paper contributes a new approach based on the theory of differential-algebraic equations. Motivated by the index reduction technique of minimal extension, a decomposition of finite element spaces is proposed that ensures stable and accurate approximations. The presented decomposition -- for standard finite element spaces used in CFD -- preserves sparsity and does not call on variable transformations which might change the meaning of the variables. Since the method is eventually an index reduction, high index effects leading to instabilities are eliminated. As a result, all constraints are maintained and one can apply semi-explicit time integration schemes.

Keywords : Navier-Stokes equations, time integration schemes, finite element method, index reduction, operator DAE

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