A finite element scheme for an entirely fractional Allen–Cahn equation with non-smooth initial data is introduced and analyzed. In the proposed nonlocal model, the Caputo fractional in-time derivative and the fractional Laplacian replace the standard local operators. Piecewise linear finite elements and convolution quadratures are the basic tools involved in the presented numerical method. Error analysis and implementation issues are addressed together with the needed results of regularity for the continuous model. Also, the asymptotic behavior of solutions, for a vanishing fractional parameter and usual derivative in time, is discussed within the framework of the -convergence theory.
Keywords: Fractional Laplacian, Caputo derivative, semilinear evolution problems
@article{M2AN_2021__55_S1_S3_0,
author = {Acosta, Gabriel and Bersetche, Francisco M.},
title = {Numerical approximations for a fully fractional {Allen{\textendash}Cahn} equation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {S3--S28},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {Suppl\'ement},
doi = {10.1051/m2an/2020022},
mrnumber = {4221301},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020022/}
}
TY - JOUR AU - Acosta, Gabriel AU - Bersetche, Francisco M. TI - Numerical approximations for a fully fractional Allen–Cahn equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - S3 EP - S28 VL - 55 IS - Supplément PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020022/ DO - 10.1051/m2an/2020022 LA - en ID - M2AN_2021__55_S1_S3_0 ER -
%0 Journal Article %A Acosta, Gabriel %A Bersetche, Francisco M. %T Numerical approximations for a fully fractional Allen–Cahn equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P S3-S28 %V 55 %N Supplément %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020022/ %R 10.1051/m2an/2020022 %G en %F M2AN_2021__55_S1_S3_0
Acosta, Gabriel; Bersetche, Francisco M. Numerical approximations for a fully fractional Allen–Cahn equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021), pp. S3-S28. doi: 10.1051/m2an/2020022
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