Kim Ngan Le

Kim Ngan Le, William McLean, Kassem Mustapha

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.

Kim Ngan Le, William McLean, Bishnu Lamichhane

An initial-boundary value problem for the time-fractional diffusion equation is discretized in space using continuous piecewise-linear finite elements on a polygonal domain with a re-entrant corner. Known error bounds for the case of a convex polygon break down because the associated Poisson equation is no longer $H^2$-regular. In particular, the method is no longer second-order accurate if quasi-uniform triangulations are used. We prove that a suitable local mesh refinement about the re-entrant corner restores second-order convergence. In this way, we generalize known results for the classical heat equation due to Chatzipantelidis, Lazarov, Thomée and Wahlbin.

Kim Ngan Le, William McLean, Kassem Mustapha

We study two schemes for a time-fractional Fokker-Planck equation with space- and time-dependent forcing in one space dimension. The first scheme is continuous in time and is discretized in space using a piecewise-linear Galerkin finite element method. The second is continuous in space and employs a time-stepping procedure similar to the classical implicit Euler method. We show that the space discretization is second-order accurate in the spatial $L_2$-norm, uniformly in time, whereas the corresponding error for the time-stepping scheme is $O(k^α)$ for a uniform time step $k$, where $α\in(1/2,1)$ is the fractional diffusion parameter. In numerical experiments using a combined, fully-discrete method, we observe convergence behaviour consistent with these results.