  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.
