Volume 54, Number 5, September-October 2020
|Page(s)||1777 - 1795|
|Published online||28 July 2020|
Higher-order finite element approximation of the dynamic Laplacian
Center for Mathematics, Technical University of Munich, Garching 85747, Germany
2 School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia
* Corresponding author: firstname.lastname@example.org
Accepted: 13 April 2020
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.
Mathematics Subject Classification: 37C30 / 37C60 / 37M99 / 65P99
Key words: Dynamic Laplacian / finite-time coherent sets / finite elements / transfer operator
© EDP Sciences, SMAI 2020
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