Finite element decomposition and minimal extension for flow equations^∗

¹ Institut für Mathematik MA4-5, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany.
raltmann@math.tu-berlin.de
² Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106 Magdeburg, Germany.

Received: 11 April 2014
Revised: 23 February 2015

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 remodelling of the flow equations is proposed. It is shown how this reformulation can be realized for standard finite elements via a decomposition of the discrete spaces and that it ensures stable and accurate approximations. The presented decomposition 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.