ISBN 978-3-8439-4453-3

978-3-8439-4453-3, Reihe Mathematik

Lucas Schöbel-Kröhn
Analysis and Numerical Approximation of Nonlinear Evolution Equations on Network Structures

147 Seiten, Dissertation Technische Universität Darmstadt (2020), Softcover, A4

Zusammenfassung / Abstract

We consider a selection of parabolic-elliptic PDE models on networks arising in different fields of study. For each of the models we establish existence and uniqueness of solutions by means of variational methods. Moreover, we systematically develop discretization schemes that preserve the most important physical properties of the underlying systems based on the finite element method.

Our first application concerns chemotaxis, the directed movement of a population induced by varying concentrations of a chemical stimulus. We propose a model with logistic population growth and prove the existence of unique global weak solutions. We then introduce a positivity-preserving finite element method and establish its convergence under general assumptions as well as convergence rates of optimal order for smooth solutions. The method is tested in numerical experiments and generic dynamics of chemotaxis systems are illustrated.

In the next example, we model the gas flow in a network of pipes by a degenerate parabolic PDE. The model is derived from the Euler equations using a simplified momentum balance. Well-posedness is proved using monotonicity and compactness arguments. The discretization is realized by a finite element method with mass lumping and the convergence of the method is obtained. As an illustration, we simulate a gas supply network with realistic model parameters.

Motivated by applications in the design of printed circuit boards, the last example is a model for the Joule heating of a domain by current flow in a network of embedded wires. We establish the well-posedness and investigate the long-time behaviour of the model. For the discretization, we propose a finite element method while asymptotic simplifications allow a fast computation time. We prove the convergence of the method and present some numerical tests.

A comprehensive appendix is included summarizing preliminaries on graphs, functional analysis, and discretization schemes.