Computational Characterization of Mixing in Flows

The major theme of this thesis is mathematical aspects of fluid mixing in the case when diffusion is negligible, which is commonly refered to as ’mixing by stirring’ or ’mixing by chaotic advection’ in the engineering literature. In this case the mixing is driven by a velocity field and is characterized by the flow generated by the velocity field. We propose a general methodology that can be used to characterize mixing in flows. In this work we assume that the velocity field is modeled by the incompressible Stokes equations but in principle we can choose to use any other fluid model. We derive pointwise a posteriori error estimates for finite element approximations of the Stokes equations and investigate the flow generated by the velocity field by computing a large number of orbits in the flow. We demonstrate that the computed orbits are close to exact orbits by deriving a shadowing error estimate. Principal to this estimate is that we compute the orbits and the velocity field sufficiently accurately. On the basis of notions from dynamical systems theory we devise a tractable mixing measure that resolves the mixing process both in space and time. We provide an error estimate for computed mixing measures which relies on the error estimate for the computed orbits. Finally, we discuss a few additional computational issues. (1) We suggest an optimal search algorithm that given a query point can locate the nsimplex in a finite element triangulation that contains the query point. (2) We analyse and discuss finite element multigrid methods for quadratic finite elements and for adaptively refined triangulations.