Eigel, M., George, E. ORCID: https://orcid.org/0000-0001-9011-3970 and Kirkilionis, M. (2010) A mesh-free partition of unity method for diffusion equations on complex domains. IMA Journal of Numerical Analysis, 30 (3). pp. 629-653. ISSN 0272-4979 (Print), 1464-3642 (Online) (doi:10.1093/imanum/drn053)

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Abstract

We present a numerical method for solving partial differential equations on domains with distinctive complicated geometrical properties. These will be called complex domains. Such domains occur in many real-world applications, for example in geology or engineering. We are, however, particularly interested in applications stemming from the life sciences, especially cell biology. In this area complex domains, such as those retrieved from microscopy images at different scales, are the norm and not the exception. Therefore geometry is expected to directly influence the physiological function of different systems, for example signalling pathways. New numerical methods that are able to tackle such problems in this important area of application are urgently needed. In particular, the mesh generation problem has imposed many restrictions in the past. The approximation approach presented here for such problems is based on a promising mesh-free Galerkin method: the partition of unity method (PUM). We introduce the main approximation features and then focus on the construction of appropriate covers as the basis of discretizations. As a main result we present an extended version of cover construction, ensuring fast convergence rates in the solution process. Parametric patches are introduced as a possible way of approximating complicated boundaries without increasing the overall problem size. Finally, the versatility, accuracy and convergence behaviour of the PUM are demonstrated in several numerical examples.

Item Type:	Article
Additional Information:	[1] Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.
Uncontrolled Keywords:	complex domains, meshfree partition of unity method, generalised finite elements, cover generation
Subjects:	Q Science > Q Science (General) Q Science > QA Mathematics
Pre-2014 Departments:	School of Computing & Mathematical Sciences
Related URLs:	Publisher
Last Modified:	23 Nov 2016 12:31
URI:	http://gala.gre.ac.uk/id/eprint/10803

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