Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation

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.

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“…Besides, Mustapha and McLean developed several discontinuous Galerkin methods [75,82,79] for a variant of the model (1.1):…”

Section: 2mentioning

confidence: 99%

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

Jin

Lazarov

Zhou

2019

Computer Methods in Applied Mechanics and Engineering

154

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“…In most situations, analytical methods do not work well on most FDEs, so the reasonable option is to resort to numerical methods. Up to now, there has been some work on numerical methods for FDEs, such as finite difference methods [12,20,26,47], finite element methods [5,6,11,14,42], spectral methods [13,15,16], and so on [28,29,30]. Numerical methods for FDEs mainly focus on the linear equations; relatively few works have been developed for the nonlinear FDEs.…”

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confidence: 99%

A Crank--Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation

Zeng¹,

Liu²,

Li³

et al. 2014

SIAM J. Numer. Anal.

330

143

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In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank-Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis. Introduction.In the past decades, fractional calculus has been used to model particle transport in porous media. Recently, there has been increasing interest in the study of fractional calculus for its wide application in many fields of science and engineering, such as the physical and chemical processes, materials, control theory, biology, finance, and so on (see [3,9,25,23,33,35]). In physics, fractional derivatives are used to model anomalous diffusion, where particles spread differently than the classical Brownian motion model [23]. Kinetic equations of the diffusion, diffusionadvection, and Fokker-Planck equations with partial fractional derivatives were recognized as a useful approach for the description of transport dynamics in complex systems. Reaction-diffusion models have been used for numerous applications in patten formation in biology, chemistry, physics, and engineering. These systems show that diffusion can produce the spontaneous formation of spatial-temporal patterns. The idea is to use a fractional-order density gradient to recover, at least at a phenomenological level, the nonhomogeneities of the porous media. Given the structural

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“…This makes the storage very expensive and challenges the algorithm design. There are several ways to discretize the time fractional derivative and speed its computation [12,19,23], McLean et al studied the convergence analysis of a discontinuous Calerkin method for the time fractional differential equation, the non-uniform time steps are used in their methods due to the singularity of derivatives at t = 0 [25,26].…”

Section: Discondnuous Galerkin Methods For Fracdonal Diffusion Equationsmentioning

confidence: 99%

High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations

Tang¹

2012

NMTMA

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As the generalization of the integer order partial differential equations (PDE), the fractional order PDEs are drawing more and more attention for their applications in fluid flow, finance and other areas. This paper presents high-order accurate Runge-Kutta local discontinuous Galerkin (DG) methods for one-and two-dimensional fractional diffusion equations containing derivatives of fractional order in space. The Gaputo derivative is chosen as the representation of spatial derivative, because it may represent the fractional derivative by an integral operator. Some numerical examples show that the convergence orders of the proposed local P'^-DG methods are 0(,h'^'^^) both in one and two dimensions, where P*" denotes the space of the real-valued polynomials with degree at most k.

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Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation

Cited by 86 publications

References 13 publications

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

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

A Crank--Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation

High-Order Accurate Runge-Kutta (Local) Discontinuous Galerkin Methods for One- and Two-Dimensional Fractional Diffusion Equations

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