We investigate a variably distributed-order time-fractional wave partial differential equation, which could accurately model, e.g., the viscoelastic behavior in vibrations in complex surroundings with uncertainties or strong heterogeneity in the data. A standard composite rectangle formula of mesh size σ is firstly used to discretize the variably distributed-order integral and then the L-1 formula of degree of freedom N is applied for the resulting fractional derivatives. Optimal error estimates of the corresponding fully-discrete finite element method are proved based only on the smoothness assumptions of the data. To maintain the accuracy, setting σ = O(N−1) leads to O(N3) operations of evaluating the temporal discretization coefficients. To improve the computational efficiency, we develop a novel time-stepping scheme by expanding the fractional kernel at a fixed fractional order to decouple the fractional operator from the variably distributed-order integral. Only O(logN) terms are needed for the expansion without loss of accuracy, which consequently reduce the computational cost of generating coefficients from O(N3) to O(N2 logN). Optimal-order error estimates of this time-stepping scheme are rigorously proved via novel and different techniques from the standard analysis procedure of the L-1 methods. Numerical experiments are presented to substantiate the theoretical results.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2021045
Keywords: Variably distributed-order time-fractional wave equation, viscoelastic problem, well-posedness and regularity, finite element method, optimal-order error estimate, fast algorithm
@article{M2AN_2021__55_5_2211_0,
author = {Jia, Jinhong and Zheng, Xiangcheng and Wang, Hong},
title = {Numerical discretization and fast approximation of a variably distributed-order fractional wave equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2211--2232},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {5},
doi = {10.1051/m2an/2021045},
mrnumber = {4323406},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021045/}
}
TY - JOUR AU - Jia, Jinhong AU - Zheng, Xiangcheng AU - Wang, Hong TI - Numerical discretization and fast approximation of a variably distributed-order fractional wave equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 2211 EP - 2232 VL - 55 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021045/ DO - 10.1051/m2an/2021045 LA - en ID - M2AN_2021__55_5_2211_0 ER -
%0 Journal Article %A Jia, Jinhong %A Zheng, Xiangcheng %A Wang, Hong %T Numerical discretization and fast approximation of a variably distributed-order fractional wave equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 2211-2232 %V 55 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021045/ %R 10.1051/m2an/2021045 %G en %F M2AN_2021__55_5_2211_0
Jia, Jinhong; Zheng, Xiangcheng; Wang, Hong. Numerical discretization and fast approximation of a variably distributed-order fractional wave equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 5, pp. 2211-2232. doi: 10.1051/m2an/2021045
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