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Higher-order finite element approximation of the dynamic Laplacian. (English) Zbl 1470.37106

Summary: 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 [G. Froyland and O. Junge, SIAM J. Appl. Dyn. Syst. 17, No. 2, 1891–1924 (2018; Zbl 1408.37139)]. 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.

MSC:

37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations

Citations:

Zbl 1408.37139
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References:

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