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IMA Journal of Numerical Analysis Advance Access published online on March 6, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn053
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

A mesh-free partition of unity method for diffusion equations on complex domains

M. Eigel{dagger}, E. George{ddagger} and M. Kirkilionis§

Mathematics Institute, University of Warwick, Zeeman Building, Coventry CV4 7AL, UK

{dagger} Email: eigel{at}maths.warwick.ac.uk

{ddagger} Email: e.george{at}warwick.ac.uk

§ Corresponding author. Email: mak{at}maths.warwick.ac.uk

Received on 14 September 2007. Revised on 29 March 2008.


  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.

Key Words: complex domains; mesh-free partition of unity method; generalized finite elements; cover generation


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