Werner Bauer

Research areas and contributions

My research contributions are in applied mathematics, in particular in numerical analysis for (geophysical) fluid dynamics (GFD), variational methods for derivations of continuous and discrete equations, in finite difference and finite element (exterior calculus) methods, in goal-oriented grid adaptation approaches for global circulation models, in differential geometric formulations of the equations of GFD, and in first contributions to stochastic fluid dynamics.

One of my overarching scientific goals is to develop consistent numerical schemes for the equations of GFD that descend from geometric structures. Inspired by similar formulations used in electrodynamics, I introducing n-dimensional split equations of GFD, split into prognostic topological and diagnostic metric equations (Bauer 2016). For this idea, I successfully acquired funding in form of a Marie Skłodowska-Curie fellowship aiming at the development of compatible finite element (FE) discretizations of the split equations of GFD. I introduced (with J. Behrens), a split FE framework for such split equations (Bauer and Behrens 2017) which we further extended (with C. Cotter) towards a structure preserving split Hamiltonian FE framework (Bauer et al. 2018). The split form mimics, on the continuous side, the structure of staggered C-grid schemes and provides new geometric insight. In particular, structure preservation of invariants (e.g. enery and enstrophy) are determined by topological prognostic equations while properties related to metric (dispersion relation, convergence) are controlled by the metric dependent diagnostic closure equations.

My interest in generalizing the split equations of GFD such that they can be derived from variational principles and to develop discretizations that follow likewise resulted in a collaboration with F. Gay-Balmaz (Ecole Normale Supérieure, Paris). We extended the variational discretization framework for incompressible (Pavlov et al. 2011) towards compressible fluids, and verified the schemes’ structure-preserving nature, such as an excellent long-term energy behavior, for various test cases on unstructured meshes (Bauer and Gay-Balmaz 2018). Currently, these variational methods are going to be used to include consistently stochastic processes into the equations of stochastic fluid dynamics.

Keywords: compatible finite element, split finite element, finite element exterior calculus, structure preserving, variational principles, Lagrangian method, Hamiltonian method, Poisson brackets, geophysical fluid dynamics, stochastic geophysical flow models, stochastic modeling, random variables, uncertainty quantification

Distinctions

• Publication promoted as a ``Featured Article'', Physics of Fluids, 10.1063/10.0007302

• Marie Skłodowska-Curie Individual Fellowship (2016- 2018) awarded by the European Union's Horizon 2020 research and innovation programme (grant agreement No 657016). Budget: 183000 Euro.

Last update: April 2023