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__Index Reduction for Semi-explicit Operator DAE's__

R. Altmann

**Abstract**

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].

**Bibliography**

[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.