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

Abstract

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order α ∈ (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.Date: May 30, 2018.

Error Analysis for a Fractional-Derivative Parabolic Problem on Quasi-Graded Meshes using Barrier Functions

“…Remark 4.9. Comparing our notation {κ m,j } with [2, (7)], the relation [2, (15)] can be rewritten as σ|κ m,m−1 | − (1 − σ)κ m,m > 0, where σ = 1 − 1 2 α, while a sufficient condition for the latter is given by [2, (17)] and is satisfied by our mesh. Furthermore, an inspection of the proof of [2,Lemma 4] shows such that in the second relation in [2, (41)], one can include a constant factor c 1 (α, σρ) ∈ (0, 1) in the right-hand side, wherē ρ := max ρ j = ρ 1 on our mesh.…”

Subdiffusion with a time-dependent coefficient: Analysis and numerical solution

“…It is of much interest to extend the analysis to high-order finite elements. This seems missing even for the case of a time-independent diffusion coefficient when problem data are nonsmooth, partly due to the limited smoothing property of the solution operators [10].…”

“…The literature on the numerical analysis of the subdiffusion problem is vast; see [21,11,9,15] for a rather incomplete list and the overview [10] (and the references therein). The work [11] analyzed two spatially semidiscrete schemes, i.e., Galerkin finite element method (FEM) and lumped mass method, and derived nearly optimal order error estimates for the homogeneous problem.…”

Numerical approximation of stochastic time-fractional diffusion

“…1473-1474], we assume the following condition: α ∈ (0, 1), γ ∈ [0, 1] and α + γ > 1/2. (1.2) The deterministic counterpart of the model (1.1), commonly known as subdiffusion, has been extensively studied in the literature over the last few decades [22], due to its numerous applications in engineering, physics and biology [31]. The noise term W (t) in the model (1.1) is to describe random effects on transport of particles in medium with memory or particles subject to sticking and trapping [11].…”

“…We employ Laplace transform and generating function [29] to derive requisite estimates. We refer interested readers to [12,26,20,21,30] for related works on nonsmooth data estimates for deterministic subdiffusion; see also the survey [22] and the references therein. For the weak convergence, we employ a powerful tool, i.e., Malliavin calculus, recently developed in [3].…”