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

• Published in: IMA journal of numerical analysis 2010-07, Vol.30 (3), p.629-653
• Main Authors: Eigel, M., George, E., Kirkilionis, M.
• Format: Article
• Language: English
• Subjects: Approximation; complex domains; Construction; cover generation; Exact sciences and technology; Functions of a complex variable; generalized finite elements; Mathematical analysis; Mathematical models; Mathematics; mesh-free partition of unity method; Meshless methods; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Partitions; Sciences and techniques of general use
• ISSN: 0272-4979; 1464-3642
• DOI: 10.1093/imanum/drn053

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.