A homotopic approach to domain determination and solution refinement for the stationary Fokker–Planck equation

An iterative approach for the solution refinement of the stationary Fokker–Planck equation is presented. The recursive use of a modified norm induced on the solution domain by the most recent estimate of the stationary probability density function, is shown to significantly improve the accuracy of the approximation over the standard L2L2-norm based Galerkin error projection. The modified norm is argued to be naturally suited to the problem, and hence preferable over the standard L2L2-norm, because the former requires substantially fewer degrees of freedom for the same order of approximation accuracy, making it immediately attractive for the Fokker–Planck equation in higher dimensions. Additionally, it is shown that the modified norm can be utilized to progress through a homotopy of dynamical systems, DpDp, in order to determine the domain of the stationary distribution of a nonlinear system of interest, (corresponding to p=1p=1) by starting with a known dynamical system (corresponding to p=0p=0) and working upwards. The partition of unity finite element method is used for numerical implementation. The meshless nature of this technique facilitates the application of the modified-norm approach to higher dimensional problems.