Adaptive finite element method assisted by stochastic simulation of chemical systems

Abstract

Stochastic models of chemical systems are often analysed 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 non-negligible 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 probability density.