A semidiscrete finite element approximation of a time-fractional Fokker-Planck equation with nonsmooth initial data
For researchers working on numerical methods for fractional PDEs, this offers a rigorous analysis for a class of problems with low-regularity data, though it is incremental as it extends existing techniques.
The paper provides a new stability and convergence analysis for the spatial discretization of a time-fractional Fokker-Planck equation using continuous, piecewise-linear finite elements, handling nonsmooth initial data and time-dependent forcing. Numerical experiments confirm the predicted convergence behavior.
We present a new stability and convergence analysis for the spatial discretization of a time-fractional Fokker--Planck equation in a convex polyhedral domain, using continuous, piecewise-linear, finite elements. The forcing may depend on time as well as on the spatial variables, and the initial data may have low regularity. Our analysis uses a novel sequence of energy arguments in combination with a generalized Gronwall inequality. Although this theory covers only the spatial discretization, we present numerical experiments with a fully discrete scheme employing a very small time step, and observe results consistent with the predicted convergence behavior.