MAT-731

In the last years numerical simulations of partial differential equations (PDEs)for biological applications became very important. The range of corresponding applications is very wide, e.g., embryonic development, cancer tumor growth, dynamic of elastic lipid membranes, vasculogenesis and angiogenesis, protein-protein interaction, tissue development and immune responses. The processes in the scope are often described with (continuum) reaction-diffusion-convection models. Very often PDEs, which are defined in a domain, have to be coupled with PDEs, which are defined on deforming-in-time surfaces. Numerical simulation of such models is a very challenging task, and modern numerical techniques are of predominant importance. In these series of lectures we start by studying the systems of chemotaxis like problems (chemotaxis = an oriented movement towards or away from regions of higher concentrations of chemical agents). We consider properties of such systems, construct a finite element numerical framework and discuss challenges, which should be taken into account while performing numerical simulations for such problems. Then, adopting the level-set method, we extend our framework to on-surface-defined PDEs. Here, the surface is implicitly prescribed by the level-set function and evolves in time according to a transport equation or minimization of an energy functional. We discuss properties of the level-set method, understand how to apply the level-set methodology for diffusion and advective terms, reconsider numerical stabilization, etc. After that we consider some biological application and discuss additional questions, such as coupling of domain- and surface-defined PDEs, and construction of methods, which allow to preserve the surface area.

The purpose of this lecture is to elaborate a finite element solver for systems of reaction-diffusion-convection equations on (evolving-in-time) surfaces/manifolds. Finite-element methods, which are studied during "Numerik für PDE" are extended by the level-set mechanisms in such a way, that one is able to treat PDEs on stationary and evolving-in-time surfaces. Flux corrected transport stabilization techniques are used to guarantee positive and non-oscillatory numerical solution. All these techniques (finite-element, level-set, FCT/TVD), when combined together, make it possible to construct an accurate and robust solver, which can be used in various real-life applications of medicine and biology.

Das Modul kann nur als benotetes Modul mit Modulprüfung abgeschlossen werden.

Zulassungsvoraussetzung für die Modulprüfung ist die Erbringung folgender Studienleistung: Regelmäßige erfolgreiche Bearbeitung der Übungsaufgaben und/oder Mitarbeit in den Übungen. Dazu kann auch eine Anwesenheitspflicht in den Übungen gehören. Details werden durch die jeweilige Dozentin / den jeweiligen Dozenten in der Veranstaltungsankündigung bekannt gemacht.

Modulprüfung: Mündliche Prüfung (ca. 30 Minuten).

Kenntnisse der Module Numerik I, II, III (Numerik für PDE) sind unabdingbar.

1. Wahlpflichtmodul für Master Mathematik, Master Technomathematik, Master Wirtschaftsmathematik
2. Angewandte Mathematik