The partition of unity finite element approach with hp-refinement for the stationary Fokker-Planck equation

In this paper, the stationary Fokker-Planck equation (FPE) is solved for nonlinear dynamical systems using a local numerical technique based on the meshless partition of unity finite element method (PUFEM). The method is applied to stationary FPE for 2-, 3- and 4-D systems and is argued to be an excellent candidate for higher dimensional problems and the transient problem. Local refinement is applied by introducing higher order polynomials in selected subdomains (local p-refinement) to keep the problem size small while ensuring high approximation accuracy. Various local approximations are blended using novel pasting functions that provide any desired order of continuity. Results are compared with existing global and local techniques. Local p-refinement is touted as an important step towards breaking the curse of dimensionality in numerical solution of FPE.