At this colloquium, we are happy to welcome:

Christian Wieners (KIT)

A parallel adaptive space-time discontinuous Galerkin method for transport in porous media

We introduce a parallel adaptive space-time discontinuous Galerkin method for the linear transport equation, where the transport vector is determined from Darcy's law for porous media flow. Given the permeability distribution, in the first step the pressure head and the flux is computed by a mixed approximation of the linear porous media problem. Then, for a given initial pollution distribution the linear transport is approximated by an adaptive DG space-time discretization on a truncated space-time cylinder which turns out to be very efficient since the adaptively refined region is transported with the
pollution distribution. The full linear system in space and time is solved with a parallel multigrid method where the stopping criterium for the linear solver is controlled by the convergence of a linear goal functional. Finally we apply this method to solve the inverse problem to reconstruct the initial pollution distribution from measurements of the outflow.

Annalaura Rebucci (MPI Leipzig)

On the commutativity of flows of two-dimensional Sobolev vector fields

A classical result in differential geometry, ultimately leading to Frobenius theorem, states that the flows of two smooth vector fields X, Y commute for all times if and only if their Lie bracket [X,Y] vanishes. In this talk, we consider continuous, Sobolev vector fields with bounded divergence on the real plan, and we discuss an extension of the classical Frobenius theorem in the setting of Regular Lagrangian Flows. In particular, we improve the previous result of Colombo-Tione (2021), where the authors require the additional assumption of the weak Lie differentiability on one of the two flows.

This is a joint project in collaboration with Martina Zizza.