In this work, we develop a novel Galerkin-L1-POD scheme for the subdiffusion model with a Caputo fractional derivative of order in time, which is often used to describe anomalous diffusion processes in heterogeneous media. The nonlocality of the fractional derivative requires storing all the solutions from time zero. The proposed scheme is based on continuous piecewise linear finite elements, L1 time stepping, and proper orthogonal decomposition (POD). By constructing an effective reduced-order model using problem-adapted basis functions, it can significantly reduce the computational complexity and storage requirement. We shall provide a complete error analysis of the scheme under realistic regularity assumptions by means of a novel energy argument. Extensive numerical experiments are presented to verify the convergence analysis and the efficiency of the proposed scheme.
Accepté le :
DOI : 10.1051/m2an/2016017
Mots clés : Fractional diffusion, energy argument, proper orthogonal decomposition, error estimates
@article{M2AN_2017__51_1_89_0,
author = {Jin, Bangti and Zhou, Zhi},
title = {An analysis of galerkin proper orthogonal decomposition for subdiffusion},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {89--113},
publisher = {EDP-Sciences},
volume = {51},
number = {1},
year = {2017},
doi = {10.1051/m2an/2016017},
mrnumber = {3601002},
zbl = {1365.65224},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2016017/}
}
TY - JOUR AU - Jin, Bangti AU - Zhou, Zhi TI - An analysis of galerkin proper orthogonal decomposition for subdiffusion JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 89 EP - 113 VL - 51 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016017/ DO - 10.1051/m2an/2016017 LA - en ID - M2AN_2017__51_1_89_0 ER -
%0 Journal Article %A Jin, Bangti %A Zhou, Zhi %T An analysis of galerkin proper orthogonal decomposition for subdiffusion %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 89-113 %V 51 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016017/ %R 10.1051/m2an/2016017 %G en %F M2AN_2017__51_1_89_0
Jin, Bangti; Zhou, Zhi. An analysis of galerkin proper orthogonal decomposition for subdiffusion. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 89-113. doi : 10.1051/m2an/2016017. http://www.numdam.org/articles/10.1051/m2an/2016017/
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