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Preprint 02-2016

Regularization and Rothe Discretization of Semi-Explicit Operator DAEs

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Author(s) : Robert Altmann , Jan Heiland

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 02-2016

MSC 2000

65J10 Equations with linear operators
65M12 Stability and convergence of numerical methods
65L80 Methods for differential-algebraic equations

Abstract :
A general framework for the regularization of constrained PDEs, also called operator differential-algebraic equations (operator DAEs), is presented. The given procedure works for semi-explicit and semi-linear operator DAEs of first order including the Navier-Stokes and other flow equations. The proposed reformulation is consistent, i.e., the solution of the PDE remains untouched. Its main advantage is that it regularizes the operator DAE in the sense that a semi-discretization in space leads to a DAE of lower index. Furthermore, a stability analysis is presented for the linear case which shows that the regularization provides benefits also for the application of the Rothe method. For this, the influence of perturbations is analyzed for the different formulations. The results are verified by means of a numerical example with an adaptive space discretization.

Keywords : operator DAE, PDAE, regularization, index reduction, Rothe method, method of lines, perturbation analysis

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