The Partition of Unity Finite Element Approach to the Stationary Fokker-Planck Equation

Abstract

The stationary Fokker-Planck Equation (FPE) is solved for nonlinear dynamic systems using a local numerical technique based on the meshless Partition of Unity Finite Element Method (PUFEM). The method is applied to the FPE for two-dimensional dynamical systems, and argued to be an excellent candidate for higher dimensional systems and the transient problem. Variations of the conventional PUFEM are used to improve the quality of approximation, by using novel pasting functions to blend the various local approximations. These functions, besides satisfying the conditions for a partition of unity are easy to integrate numerically and provide solution continuity of any desired order. Results are compared with existing global and local techniques.