Adaptive Finite Element Method Assisted by Stochastic Simulation of Chemical Systems

Cotter, Simon L.; Vejchodský, Tomáš; Erban, Radek

Adaptive Finite Element Method Assisted by Stochastic Simulation of Chemical Systems

Type

Article

Authors

Cotter, Simon L.
Vejchodský, Tomáš
Erban, Radek

KAUST Grant Number

KUK-C1-013-04

Date

2013-01

Abstract

Stochastic models of chemical systems are often analyzed by solving the corresponding Fokker-Planck equation, which is a drift-diffusion partial differential equation for the probability distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with nonnegligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the stationary probability density. Numerical examples demonstrate that the presented method is competitive with existing a posteriori methods. © 2013 Society for Industrial and Applied Mathematics.

Citation

Cotter SL, Vejchodský T, Erban R (2013) Adaptive Finite Element Method Assisted by Stochastic Simulation of Chemical Systems. SIAM Journal on Scientific Computing 35: B107–B131. Available: http://dx.doi.org/10.1137/120877374.

Acknowledgements

Submitted to the journal's Computational Methods in Science and Engineering section May 15, 2012; accepted for publication (in revised form) December 3, 2012; published electronically January 10, 2013. This work was supported by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement 239870 and was based on work supported in part by award KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom (simon.cotter@manchester.ac.uk). This author's work was partially supported by a Junior Research Fellowship of St Cross College, University of Oxford.Institute of Mathematics, Czech Academy of Sciences, Zitna 25, 115 67 Praha 1, Czech Republic (vejchod@math.cas.cz). This author's work was supported by the Grant Agency of the Academy of Sciences (project IAA100190803) and RVO 67985840.Mathematical Institute, University of Oxford, 24-29 St. Giles', Oxford, OX1 3LB, United Kingdom (erban@maths.ox.ac.uk). This author's work was supported by Somerville College, University of Oxford, by a Fulford Junior Research Fellowship; Brasenose College, University of Oxford, by a Nicholas Kurti Junior Fellowship; the Royal Society for a University Research Fellowship; and the Leverhulme Trust for a Philip Leverhulme Prize. This prize money was used to support research visits of Tomas Vejchodsky in Oxford.