An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations

Type
Article

Authors
Burrage, Kevin
Hale, Nicholas
Kay, David

KAUST Grant Number
KUK-C1-013-04

Date
2012-01

Abstract
Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that 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 via 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 analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator. © 2012 Society for Industrial and Applied Mathematics.

Citation
Burrage K, Hale N, Kay D (2012) An Efficient Implicit FEM Scheme for Fractional-in-Space Reaction-Diffusion Equations. SIAM Journal on Scientific Computing 34: A2145–A2172. Available: http://dx.doi.org/10.1137/110847007.

Acknowledgements
This author's work was supported by award KUK-C1-013-04 from King Abdullah University of Science and Technology (KAUST).

Publisher
Society for Industrial & Applied Mathematics (SIAM)

Journal
SIAM Journal on Scientific Computing

DOI
10.1137/110847007

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