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Vorlesung

Numerical simulation of PDEs for biological applications

Nummer
011286, WS1314
Dozentinnen und Dozenten
Veranstaltungstyp
Vorlesung, 2+1
Ort und Zeit
M/1011 Di 14:00 2h
Modul-Zugehörigkeit (ohne Gewähr)
DPL:B:-:2 – Mathematik, Diplom (auslaufend)
DPL:E:-:- – Mathematik, Promotionsstudiengang
MAMA:-:7:MAT-731 – Numerical simulations of PDEs for biological applications
WIMAMA:-:7:MAT-731 – Numerical simulations of PDEs for biological applications
TMAMA:-:7:MAT-731 – Numerical simulations of PDEs for biological applications
Inhalt

In the last years numerical simulations of partial diff erential 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 diff usion 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.

Empfohlene Literatur
  • Stanley Osher, Roland Fedkiw: Level Set methods and dynamic implicit surfaces, Springer-Verlag New York, Inc. 2003
  • Gerhard Dziuk, Charles M. Elliot: Finite Element methods for surface PDEs, Acta Numerica 22, pp. 289-336, 2013
  • Dirk Horstmann: From 1970 until present: The Keller-Segel model in chemotaxis and its consequences I, Jahresbericht der DMV 105, no. 3, pp. 103-165, 2003

Übungen

Nummer der Übung
011287
Übungsgruppen
n.V.