A mixed FEM for a time-fractional Fokker-Planck model
Samir Karaa, Kassem Mustapha, Naveed Ahmed
TL;DR
The paper addresses the numerical solution of a time-fractional Fokker-Planck equation with a space-dependent drift using a mixed finite element discretization. It develops a semidiscrete mixed FEM, establishes $\alpha$-robust, optimal convergence for both the primary variable $u$ and the flux $\boldsymbol\sigma$ via an energy-based error-splitting approach, and couples the spatial discretization with a backward-Euler convolution quadrature to obtain pointwise-in-time $L^2$-error bounds. The fully discrete analysis yields sharp, time-graded $L^2$-error estimates for the primary variable, validated by numerical experiments that show expected spatial and temporal convergence rates for smooth and nonsmooth initial data. Overall, the work provides a robust, alpha-robust numerical framework for time-fractional diffusion with drift and contributes to error analysis techniques for fractional PDEs.
Abstract
We propose and analyze a mixed finite element method for the spatial approximation of a time-fractional Fokker--Planck equation in a convex polyhedral domain, where the given driving force is a function of space. Taking into account the limited smoothing properties of the model, and considering an appropriate splitting of the errors, we employed a sequence of clever energy arguments to show optimal convergence rates with respect to both approximation properties and regularity results. In particular, error bounds for both primary and secondary variables are derived in $L^2$-norm for cases with smooth and nonsmooth initial data. We further investigate a fully implicit time-stepping scheme based on a convolution quadrature in time generated by the backward Euler method. Our main result provides pointwise-in-time optimal $L^2$-error estimates for the primary variable. Numerical examples are then presented to illustrate the theoretical contributions.