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.
Mots-clés : 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 },
pages = {1845--1864},
publisher = {EDP-Sciences},
volume = {47},
number = {6},
year = {2013},
doi = {10.1051/m2an/2013091},
mrnumber = {3123379},
zbl = {1282.35400},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2013091/}
}
TY - JOUR AU - Deng, W. H. AU - Hesthaven, J. S. TI - Local Discontinuous Galerkin methods for fractional diffusion equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 1845 EP - 1864 VL - 47 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2013091/ DO - 10.1051/m2an/2013091 LA - en ID - M2AN_2013__47_6_1845_0 ER -
%0 Journal Article %A Deng, W. H. %A Hesthaven, J. S. %T Local Discontinuous Galerkin methods for fractional diffusion equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1845-1864 %V 47 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2013091/ %R 10.1051/m2an/2013091 %G en %F 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 , Tome 47 (2013) no. 6, pp. 1845-1864. doi : 10.1051/m2an/2013091. http://archive.numdam.org/articles/10.1051/m2an/2013091/
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