Hybrid stochastic finite element method for solving Fokker-Planck equations

When modelling biochemical reactions within cells, it is vitally important to take into account the effect of intrinsic noise in the system, due to the small copy numbers of some of the chemical species. Deterministic systems can give vastly different types of behaviour for the same parameter sets of reaction rates as their stochastic analogues, giving us an incorrect view of the bifurcation behaviour.

The stochastic description of this problem gives rise to a multi-dimensional Markov jump process, which can be approximated by a system of stochastic differential equations. Long-time behaviour of the process can be better understood by looking at the steady-state solution of the corresponding Fokker-Planck equation.

In this talk we consider a new finite element method which uses simulated trajectories of the Markov-jump process to inform the choice of mesh in order to approximate this invariant distribution. The method has been implemented for systems in 3 dimensions, but we shall also consider systems of higher dimension.