A Discontinuous Petrov--Galerkin Method for Time-Fractional Diffusion Equations

We present optimal error estimates for spectral Petrov-Galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval. We also develop Laguerre spectral Petrov-Galerkin methods and collocation methods for fractional equations on the half line. Numerical results confirm the error estimates. Introduction.Numerical methods for fractional differential equations have been investigated for decades; see, e.g., [8,12,33]. However, spectral methods for these equations have a rather short history since the solutions usually have some singular lower-order derivatives. The recent investigation of spectral methods for these singular problems is motivated by the following facts. First, all numerical methods for these problems are nonlocal as spectral methods are. For example, in finite difference methods, see, e.g., [33], and in finite element methods, see, e.g., [13,18], data at almost all grids or all elements are exploited to approximate the integral operators at one grid or element. These methods are still nonlocal, even when a "fixed memory principle" is applied to reduce the use of all data; see, e.g., [15,19]. Second, spectral methods lead to high-order accuracy when the solutions are smooth; see, e.g., [7,24] for integerorder differential equations. For fractional differential equations with weakly singular kernels, solutions can be smooth even when some inputs of the equations are singular.Consider the following fractional ordinary differential equation (FODE) over the interval I = (−1, 1):

Section: Introductionmentioning

confidence: 99%

Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

Zhang¹,

Zeng²,

Karniadakis³

2015

“…The piecewise linear case has a local truncation error O(τ 2−α ) for sufficiently smooth solution, where τ denotes the time step size. See also [31,33] for the discontinuous Galerkin method. CQ is a flexible framework introduced by Lubich [27,28] for constructing high-order time discretization methods for approximating fractional derivatives.…”

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confidence: 99%

Numerical Analysis of Nonlinear Subdiffusion Equations

Jin¹,

Li²,

Zhou³

2018

203

133

Abstract. We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order α ∈ (0, 1) in time. It relies on three technical tools: a fractional version of the discrete Grönwall-type inequality, discrete maximal regularity, and regularity theory of nonlinear equations. We establish a general criterion for showing the fractional discrete Grönwall inequality, and verify it for the L1 scheme and convolution quadrature generated by BDFs. Further, we provide a complete solution theory, e.g., existence, uniqueness and regularity, for a time-fractional diffusion equation with a Lipschitz nonlinear source term. Together with the known results of discrete maximal regularity, we derive pointwise L 2 (Ω) norm error estimates for semidiscrete Galerkin finite element solutions and fully discrete solutions, which are of order O(h 2 ) (up to a logarithmic factor) and O(τ α ), respectively, without any extra regularity assumption on the solution or compatibility condition on the problem data. The sharpness of the convergence rates is supported by the numerical experiments.Keywords: nonlinear fractional diffusion equation, discrete fractional Grönwall inequality, L1 scheme, convolution quadrature, error estimate 1. Introduction. Time-fractional parabolic partial differential equations (PDEs) have been very popular for modeling anomalously slow transport processes in the past two decades. These models are commonly referred to as fractional diffusion or subdiffusion. At a microscopic level, the underlying stochastic process is continuous time random walk [32]. So far they have been successfully applied in a broad range of diversified research areas, e.g., thermal diffusion in fractal domains [35], flow in highly heterogeneous aquifer [6] and single-molecular protein dynamics [20], just to name a few. Hence, the rigorous numerical analysis of such problems is of great practical importance. For the linear problem, various efficient time stepping schemes have been proposed, which include mainly two classes: L1 type schemes and convolution quadrature (CQ).L1 type schemes approximate the fractional derivative by replacing the integrand with its piecewise polynomial interpolation [24,26,37,3] and thus generalize the classical finite difference method. The piecewise linear case has a local truncation error O(τ 2−α ) for sufficiently smooth solution, where τ denotes the time step size. See also [31,33] for the discontinuous Galerkin method. CQ is a flexible framework introduced by Lubich [27,28] for constructing high-order time discretization methods for approximating fractional derivatives. It approximates the fractional derivative in the Laplace domain and automatically inherits the stability property of general linear multistep methods. See [10,39,40,16] for CQ type schemes. Optimal error estimates have been derived for both spatially semidiscrete and fully discrete schemes, including problems with nonsmooth data [10,14,31,16]....

“…There has been much recent interest in developing numerical methods for (1.1), especially spectral methods, [4], [5], [43], [45], and the discontinuous Galerkin method [8], [32], [33], [34]. In this paper, we will consider some time discretization schemes for (1.1) using the direct approximation of the time fractional derivative.…”

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confidence: 99%

An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data

Yan¹,

Khan²,

Ford³

2018

151

Abstract. We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin et al. (2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. of Numer. Anal.,36, established an O(k) convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where k denotes the time step size. We show that the modified L1 scheme has convergence rate O(k 2−α ), 0 < α < 1 for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results.