# Robust FEM-based extraction of finite-time coherent sets using scattered, sparse, and incomplete trajectories

## TL;DR

This paper introduces FEM-based numerical methods to identify finite-time coherent sets in flows using limited, scattered observational data, aiding the analysis of transport and mixing in complex systems.

## Contribution

The paper develops three FEM-based algorithms for approximating the dynamic Laplace operator and introduces a new dynamic isoperimetric problem with Dirichlet conditions for better coherent set detection.

## Key findings

- Efficient extraction of finite-time coherent sets from sparse data.
- Reliable identification of flow regions with minimal mixing.
- Applicable to real-world geophysical flow data.

## Abstract

Transport and mixing properties of aperiodic flows are crucial to a dynamical analysis of the flow, and often have to be carried out with limited information. Finite-time coherent sets are regions of the flow that minimally mix with the remainder of the flow domain over the finite period of time considered. In the purely advective setting this is equivalent to identifying sets whose boundary interfaces remain small throughout their finite-time evolution. Finite-time coherent sets thus provide a skeleton of distinct regions around which more turbulent flow occurs. They manifest in geophysical systems in the forms of e.g.\ ocean eddies, ocean gyres, and atmospheric vortices. In real-world settings, often observational data is scattered and sparse, which makes the difficult problem of coherent set identification and tracking even more challenging. We develop three FEM-based numerical methods to efficiently approximate the dynamic Laplace operator, and introduce a new dynamic isoperimetric problem using Dirichlet boundary conditions. Using these FEM-based methods we rapidly and reliably extract finite-time coherent sets from models or scattered, possibly sparse, and possibly incomplete observed data.