Local Discontinuous Galerkin methods for fractional diffusion equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 6, p. 1845-1864

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

DOI : https://doi.org/10.1051/m2an/2013091
Classification:  35R11,  65M60,  65M12
Keywords: fractional derivatives, local discontinuous Galerkin methods, stability, convergence, error estimates
@article{M2AN_2013__47_6_1845_0,
     author = {Deng, W. H. and Hesthaven, J. S.},
     title = {Local Discontinuous Galerkin methods for fractional diffusion equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {6},
     year = {2013},
     pages = {1845-1864},
     doi = {10.1051/m2an/2013091},
     zbl = {1282.35400},
     mrnumber = {3123379},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2013__47_6_1845_0}
}
Deng, W. H.; Hesthaven, J. S. Local Discontinuous Galerkin methods for fractional diffusion equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 6, pp. 1845-1864. doi : 10.1051/m2an/2013091. http://www.numdam.org/item/M2AN_2013__47_6_1845_0/

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