The dynamic Laplace operator arises from extending problems of isoperimetry from fixed manifolds to manifolds evolved by general nonlinear dynamics. Eigenfunctions of this operator are used to identify and track finite-time coherent sets, which physically manifest in fluid flows as jets, vortices, and more complicated structures. Two robust and efficient finite-element discretisation schemes for numerically computing the dynamic Laplacian were proposed in Froyland and Junge [SIAM J. Appl. Dyn. Syst. 17 (2018) 1891–1924]. In this work we consider higher-order versions of these two numerical schemes and analyse them experimentally. We also prove the numerically computed eigenvalues and eigenvectors converge to the true objects for both schemes under certain assumptions. We provide an efficient implementation of the higher-order element schemes in an accompanying Julia package.
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DOI : 10.1051/m2an/2020027
Mots-clés : Dynamic Laplacian, finite-time coherent sets, finite elements, transfer operator
@article{M2AN_2020__54_5_1777_0,
author = {Schilling, Nathanael and Froyland, Gary and Junge, Oliver},
title = {Higher-order finite element approximation of the dynamic {Laplacian}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1777--1795},
publisher = {EDP-Sciences},
volume = {54},
number = {5},
year = {2020},
doi = {10.1051/m2an/2020027},
mrnumber = {4127955},
zbl = {1470.37106},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2020027/}
}
TY - JOUR AU - Schilling, Nathanael AU - Froyland, Gary AU - Junge, Oliver TI - Higher-order finite element approximation of the dynamic Laplacian JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2020 SP - 1777 EP - 1795 VL - 54 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2020027/ DO - 10.1051/m2an/2020027 LA - en ID - M2AN_2020__54_5_1777_0 ER -
%0 Journal Article %A Schilling, Nathanael %A Froyland, Gary %A Junge, Oliver %T Higher-order finite element approximation of the dynamic Laplacian %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2020 %P 1777-1795 %V 54 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2020027/ %R 10.1051/m2an/2020027 %G en %F M2AN_2020__54_5_1777_0
Schilling, Nathanael; Froyland, Gary; Junge, Oliver. Higher-order finite element approximation of the dynamic Laplacian. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 5, pp. 1777-1795. doi : 10.1051/m2an/2020027. http://www.numdam.org/articles/10.1051/m2an/2020027/
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