Dissertation/Thesis Abstract

Regularization and Simulation of Constrained Partial Differential Equations
by Altmann, Robert, Dr.Nat., Technische Universitaet Berlin (Germany), 2015, 147; 10699740
Abstract (Summary)

This thesis is devoted to the regularization of differential-algebraic equations in the abstract setting (operator DAEs) and the resulting positive impact on the corresponding semi-discrete systems and on the sensitivity to perturbations. The possibility of a modularized modeling and the maintenance of the physical structure of a dynamical system make operator DAEs convenient form the modeling point of view. They appear in all fields of applications such as fluid dynamics, elastodynamics, electromagnetics, as well as in multi-physics applications were different system types are coupled. From a mathematical point of view, operator DAEs are constrained PDEs, written in the weak formulation. Therein, the constraint may itself be a differential equation such as in the Navier-Stokes equations where the velocity of a Newtonian fluid is constrained to be divergence-free. On the other hand, operator DAEs generalize the notion of DAEs to the infinite-dimensional setting, including abstract functions which map into a Banach space. Thus, a spatial discretization leads to a DAE in the classical sense. This also implies that typical stability issues known from the theory of DAEs such as the high sensitivity to perturbations also translate to the operator case. The regularization of an operator DAE follows the concept of an index reduction for a DAE. Hence, an equivalent system is sought-after which has better properties from a numerical point of view. The presented regularization lifts the index reduction technique of minimal extension for semi-explicit DAEs to the abstract setting and leads to an extended operator DAE. A spatial discretization of the regularized system then leads to a DAE of lower index compared to the semi-discrete system arising from the original operator DAE. For flow equations we obtain a reduction from index 2 to index 1 whereas the applications from the field of elastodynamics yield a reduction from index 3 to index 1. The last part of this thesis deals with the convergence of time discretization schemes applied to the regularized operator DAEs. Therein, we observe a qualitative difference for different variables. More precisely, we show that the Lagrange multiplier needs stronger regularity assumptions on the given data in order to guarantee the convergence to the exact solution of the operator DAE. Furthermore, the influence of perturbations in the right-hand sides of the system is analysed for the semi-discrete as well as for the continuous setting. This analysis shows the advantage of the presented regularization in terms of stability.

Indexing (document details)
Advisor: Mehrmann, Volker
School: Technische Universitaet Berlin (Germany)
School Location: Germany
Source: DAI-C 81/1(E), Dissertation Abstracts International
Subjects: Applied Mathematics
Keywords: Differential algebraic equations
Publication Number: 10699740
ISBN: 9781392870167
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