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Abstract: In this paper, a semianalytic partition of unity finite element method (PUFEM) is presented to solve the transient Fokker–Planck equation (FPE) for high-dimensional nonlinear dynamical systems. Meshless spatial discretization of PUFEM with local -refinement (discussed in a previous paper) is employed to develop linear ordinary differential equations for the time varying coefficients of local shape functions. A similarity transformation to modal coordinates is shown to reveal numerous spurious modes in the eigenspace of the discretized FPE operator. Identification and elimination of these modes leads to an analytical solution of ODEs obtained from spatial discretization in terms of the remaining admissible modes, and a significant reduction in the size of the discretized transient problem. Initial equation error resulting from the set of admissible modes is shown to be an upper bound for all time, meaning that the reduced admissible set is sufficient for the FPE approximation for all time. [Copyright &y& Elsevier]

Abstract: 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 -norm based Galerkin error projection. The modified norm is argued to be naturally suited to the problem, and hence preferable over the standard -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, , in order to determine the domain of the stationary distribution of a nonlinear system of interest, (corresponding to ) by starting with a known dynamical system (corresponding to ) 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. [Copyright &y& Elsevier]

Abstract: 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 -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 -refinement is touted as an important step towards breaking the curse of dimensionality in numerical solution of FPE. [Copyright &y& Elsevier]