SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Contributed Talk

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Index Reduction for Semi-explicit Operator DAE's

R. Altmann

The talk is devoted to semi-explicit operator differential equations of the form $ \dot u(t) + K u(t) + B^* \lambda (t) &= F(t), B u(t) \phantom{j+Bu(t)} &= G(t) $ with given initial value. Because of the saddle point structure, standard semi-discretization schemes in space such as finite elements lead to a differential-algebraic equation (DAE) of index $2$. Thus, we call the above equation an operator DAE. An example of a system with this structure is given by the Navier-Stokes equations, where $B$ equals the divergence operator and $\lambda$ stands for the pressure. For a certain class of constraint operators $B$, we present a reformulation of the above system, which can be seen as an index reduction procedure on operator level. By this we mean that a semi-discretization of the reformulated system gives a DAE of index $1$. The presented method is based on the index reduction technique of minimal extension [1].

[1] P. Kunkel and V. Mehrmann. Index reduction for differential-algebraic equations by minimal extension, Z. Angew. Math. Mech. (ZAMM), 84(9) 2004, pp. 579-597.

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