Eigel, M., George, E. and Kirkilionis, Markus (2010) A mesh-free partition of unity method for diffusion equations on complex domains. IMA Journal of Numerical Analysis, Vol.30 (No.3). pp. 629-653. doi:10.1093/imanum/drn053 ISSN 0272-4979.

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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:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	IMA Journal of Numerical Analysis
Publisher:	Oxford University Press
ISSN:	0272-4979
Official Date:	July 2010
Dates:	Date Event July 2010 Published
Volume:	Vol.30
Number:	No.3
Number of Pages:	25
Page Range:	pp. 629-653
DOI:	10.1093/imanum/drn053
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	European Commission, Engineering and Physical Sciences Research Council (EPSRC), Biotechnology and Biological Sciences Research Council (Great Britain) (BBSRC), University of Warwick. Centre for Scientific Computing
Grant number:	12990, BB/C00437X/1 (BBSRC)

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