Estimation of Variable Cefficients in the Fokker–Planck Quations Using Moving Node Finite Elements

Inverse problems are considered for the estimation of temporally and spatially varying coefficients in the Fokker–Planck or forward Kolmogorov equation that often arise in size/age structured population models. These are difficult problems even for simulation studies since they share certain numerical difficulties with transport-dominated diffusion-convection problems of fluid dynamics. These difficulties stem from the fact that when the convection is large compared to the diffusion, both traditional finite difference or finite element methods produce erroneous oscillatory solutions. In this paper, computational techniques are presented that combine a variation of the moving finite element method with spline approximations for the parameter estimation problems. Computational details of our numerical algorithm along with theoretical convergence results are presented. Several numerical examples illustrating the effectiveness of the method are also given.