# Higher-Order Finite Element Approximation of the Dynamic Laplacian

## TL;DR

This paper develops and analyzes higher-order finite element methods for approximating the dynamic Laplacian, which helps identify coherent structures in fluid flows, with proven convergence and an accompanying Julia implementation.

## Contribution

It introduces higher-order discretization schemes for the dynamic Laplacian and proves their convergence, enhancing numerical accuracy for identifying coherent structures.

## Key findings

- Higher-order schemes improve approximation accuracy.
- Eigenvalues and eigenvectors converge under certain conditions.
- Efficient Julia implementation is provided.

## Abstract

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 & Junge (2018). 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.