Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following aspects of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space-time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.Comment: 24 pages, 3 figure

Local discontinuous Galerkin methods for fractional ordinary differential equations

This paper discusses the upwinded local discontinuous Galerkin methods for the one-term/multi-term fractional ordinary differential equations (FODEs). The natural upwind choice of the numerical fluxes for the initial value problem for FODEs ensures stability of the methods. The solution can be computed element by element with optimal order of convergence $k+1$ in the $L^2$ norm and superconvergence of order $k+1+\min\{k,\alpha\}$ at the downwind point of each element. Here $k$ is the degree of the approximation polynomial used in an element and $\alpha$ ($\alpha\in (0,1]$) represents the order of the one-term FODEs. A generalization of this includes problems with classic $m$'th-term FODEs, yielding superconvergence order at downwind point as $k+1+\min\{k,\max\{\alpha,m\}\}$. The underlying mechanism of the superconvergence is discussed and the analysis confirmed through examples, including a discussion of how to use the scheme as an efficient way to evaluate the generalized Mittag-Leffler function and solutions to more generalized FODE's.Comment: 17 pages, 7 figure

The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation

We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one and two-dimension problems confirm the convergence rates of the theoretical results.Comment: 22 pages, 4 figure

Correction of high-order BDF convolution quadrature for fractional evolution equations

We develop proper correction formulas at the starting $k-1$ steps to restore the desired $k^{\rm th}$-order convergence rate of the $k$-step BDF convolution quadrature for discretizing evolution equations involving a fractional-order derivative in time. The desired $k^{\rm th}$-order convergence rate can be achieved even if the source term is not compatible with the initial data, which is allowed to be nonsmooth. We provide complete error estimates for the subdiffusion case $\alpha\in (0,1)$, and sketch the proof for the diffusion-wave case $\alpha\in(1,2)$. Extensive numerical examples are provided to illustrate the effectiveness of the proposed scheme.Comment: 22 pages, 3 figure