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.
DOI : 10.1051/m2an/2015029
Mots-clés : Navier−Stokes equations, time integration schemes, finite element method, index reduction, operator DAE
@article{M2AN_2015__49_5_1489_0, author = {Altmann, R. and Heiland, J.}, title = {Finite element decomposition and minimal extension for flow equations}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1489--1509}, publisher = {EDP-Sciences}, volume = {49}, number = {5}, year = {2015}, doi = {10.1051/m2an/2015029}, zbl = {1327.76090}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an/2015029/} }
TY - JOUR AU - Altmann, R. AU - Heiland, J. TI - Finite element decomposition and minimal extension for flow equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2015 SP - 1489 EP - 1509 VL - 49 IS - 5 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2015029/ DO - 10.1051/m2an/2015029 LA - en ID - M2AN_2015__49_5_1489_0 ER -
%0 Journal Article %A Altmann, R. %A Heiland, J. %T Finite element decomposition and minimal extension for flow equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2015 %P 1489-1509 %V 49 %N 5 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2015029/ %R 10.1051/m2an/2015029 %G en %F M2AN_2015__49_5_1489_0
Altmann, R.; Heiland, J. Finite element decomposition and minimal extension for flow equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1489-1509. doi : 10.1051/m2an/2015029. http://archive.numdam.org/articles/10.1051/m2an/2015029/
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