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Title: An adaptive meshfree method for reaction-diffusion processes on complex domains applications in cell biology
Author: Eigel, Martin
ISNI:       0000 0004 2672 3530
Awarding Body: The University of Warwick
Current Institution: University of Warwick
Date of Award: 2008
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A mesh-free numerical method for solving partial differential equations (PDE) on geometrically complex domains is presented. The approach is based on the general Partition of Unity Method (PUM), a framework developed by Babuska and Melenk in [8, 122]. In order to underline the universality of our method, we summarise the current state-of-the-art of the PUM in a comprehensive, self-contained way. Additionally, as an interesting research direction, fundamental and recent a posteriori error estimation techniques are discussed and transferred to the PUM. These are employed to steer an adaptive refinement process with the aim to reduce the overall error in an efficient way, Le. with the least additional computational complexity. Particular attention is paid to the targeted application area: the life sciences and, more specifically, transport processes in cell biology. Biological environments exhibit several features which make them quite different to classical fields of application. We discuss some of these properties and propose a novel approach to define and examine complex transport mechanisms based on 'elementary modules' describing typical phenomena. This systematic approach culminates in the definition of a biological transport toolbox whose building blocks can be plugged together to form more complex systems. One of the paramount challenges one has to tackle in biological simulations is the spatial discretisation of c'omplex geometries. Hence, we develop a general method to automatically handle complicated shapes which may be organised hierarchically, thus allowing for the simulation of processes spanning several nested domains. Additionally, processes may also take place on the interfaces, reminiscent of biological binding and transport phenomena on membranes. The numerical technique employed in this case is a mixed mesh-based and mesh-free discretisation. '!\vo interesting and important biological systems are discussed in order to relate the presented ideas to real-world applications. First, we summarise some properties of the nucleus of the eukaryote cell, a compartment where nearly the entire genetic information of a cell is contained. Second, the thylakoid, an organelle of the chloroplast of the plant cell, is depicted. Typical transport processes encountered in these model systems are then examined to demonstrate the developed methodology. The powerful combination of a systematic and modular modelling approach and 'in silica' experiments based on a versatile numerical method represents, we hope, an important contribution to the interdisciplinary research required to attack the multitude of challenges the study of biological systems poses.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available