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

Kevin Burrage, Nicholas Hale, David Kay

Research output: Contribution to journalArticlepeer-review

145 Scopus citations

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.
Original languageEnglish (US)
Pages (from-to)A2145-A2172
Number of pages1
JournalSIAM Journal on Scientific Computing
Volume34
Issue number4
DOIs
StatePublished - Jan 2012
Externally publishedYes

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