Adaptive Finite Element Method Assisted by Stochastic Simulation of Chemical Systems

TL;DR

This paper introduces an adaptive finite element method that leverages short-term stochastic simulations to efficiently refine meshes for solving the Fokker-Planck equation in chemical systems, improving computational accuracy.

Contribution

The novel approach combines stochastic simulation with adaptive mesh refinement to enhance numerical solutions of the Fokker-Planck equation for chemical systems.

Findings

Efficient identification of high-probability regions via stochastic trajectories.

Improved accuracy in probability density computation with adaptive meshes.

Reduction in computational cost compared to uniform meshing.

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.