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Local Discontinuous Galerkin methods for fractional diffusionequations∗∗∗

Published online by Cambridge University Press:  07 October 2013

W.H. Deng  and
J.S. Hesthaven
W.H. Deng
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China; Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA.. dengwh@lzu.edu.cn
J.S. Hesthaven
Affiliation:
Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA.; Jan.Hesthaven@Brown.edu
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Abstract

We consider the development and analysis of local discontinuous Galerkin methods forfractional diffusion problems in one space dimension, characterized by having fractionalderivatives, parameterized by β ∈[1, 2]. After demonstrating that aclassic approach fails to deliver optimal order of convergence, we introduce a modifiedlocal numerical flux which exhibits optimal order of convergence 𝒪(hk + 1) uniformly across the continuous range between pureadvection (β = 1) and pure diffusion (β = 2). In the twoclassic limits, known schemes are recovered. We discuss stability and present an erroranalysis for the space semi-discretized scheme, which is supported through a fewexamples.

Keywords

Fractional derivativeslocal discontinuous Galerkin methodsstabilityconvergenceerror estimates
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013 , pp. 1845 - 1864
Copyright
© EDP Sciences, SMAI 2013

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