The Mathematical Institute, University of Oxford, Eprints Archive

An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations

Burrage, K. and Hale, Nicholas and Kay, David (2011) An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations. Technical Report. SIAM. (Submitted)

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Abstract

Fractional differential equations are becoming increasingly used as a modelling tool for processes with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues, which impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids, and robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analysing the speed of the travelling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator.

Item Type:Technical Report (Technical Report)
Subjects:H - N > Numerical analysis
Research Groups:Oxford Centre for Collaborative Applied Mathematics
Numerical Analysis Group
ID Code:1358
Deposited By:Lotti Ekert
Deposited On:04 Aug 2011 14:25
Last Modified:09 Feb 2012 13:48

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