Abstract.In the simulation of flows, the correct treatment of the pressure variable is the key to stabletime-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 remodellingof the flow equations is proposed. It is shown how this reformulation can be realized for standard finiteelementsviaa decomposition of the discrete spaces and that it ensures stable and accurate approxi-mations. The presented decomposition preserves sparsity and does not call on variable transformationswhich might change the meaning of the variables. Since the method is eventually an index reduction,high index effects leading to instabilities are eliminated.