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The partition of unity meshfree method for solving transport-reaction equations on complex domains: implementation and applications in the life sciences

The partition of unity meshfree method for solving transport-reaction equations on complex domains: implementation and applications in the life sciences

Eigel, Martin, George, Erwin ORCID: 0000-0001-9011-3970 and Kirkilionis, Markus (2008) The partition of unity meshfree method for solving transport-reaction equations on complex domains: implementation and applications in the life sciences. In: Griebel, Michael and Schweitzer, Marc Alexander, (eds.) Meshfree Methods for Partial Differential Equations IV. Lecture Notes in Computational Science and Engineering (65). Springer, Berlin Heidelberg, pp. 69-93. ISBN 9783540799931 (doi:https://doi.org/10.1007/978-3-540-79994-8_5)

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Abstract

There is a wide range of highly significant scientific problems which on appropriate physical scales can be formulated as partial differential equations defined on so-called complex domains. Such complex domains often occur when material is transported through an environment of high geometrical complexity, for example porous media, domains with many obstacles, or membrane systems that are folded in a topologically complex configuration. The latter often occurs in cell biology, where the biological membranes inside the cell are strikingly topologically complex. In addition the medium in which, for example, proteins diffuse in the cell nucleus, is a complex porous media type of environment as many macro-molecules and protein-DNA complexes like the chromatin form a highly irregular structure in which many bio-molecular interactions occur. The distribution of biomolecules inside cells and tissues, their over-abundance or absence in metabolism, signalling etc., is the cause of many human diseases, therefore numerical simulations will be essential for future diagnostic abilities. Under appropriate assumptions the resulting molecular transport can be formulated as a PDE (Partial Differential Equation). The first challenge for any numerical discretisation is the generation of a cover for the underlying computational domain. Here, the meshfree Partition of Unity Method (PUM) offers a number of new degrees of freedom, as patches can be shifted, their size increased or diminished, with no need to create a non-overlapping cover at all times as is characteristic for traditional Finite Element and Finite Volume discretisations. Further advances in cover creation algorithms as discussed in this paper will allow the routine simulation of problems on domains with more complex
geometries than have been treatable before.

Item Type: Book Section
Additional Information: [1] ISBN: 9783540799931 (print); 9783540799948 (online) [2] ISSN: 1439-7358
Uncontrolled Keywords: complex domains, Partition of Unity Meshfree, PUM, meshfree discretisation, cover construction, cell biology
Subjects: Q Science > Q Science (General)
Q Science > QA Mathematics > QA76 Computer software
Pre-2014 Departments: School of Computing & Mathematical Sciences
Related URLs:
Last Modified: 14 Oct 2016 09:26
Selected for GREAT 2016: None
Selected for GREAT 2017: None
Selected for GREAT 2018: None
URI: http://gala.gre.ac.uk/id/eprint/10759

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