A semianalytic meshless approach to the transient Fokker–Planck equation

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 pp-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.