Past Event: Oden Institute Seminar

On the role of the Helmholtz-Hodge projector for a novel pressure-robust discretization theory for the incompressible Navier-Stokes equations

Alexander Linke, Numerical Mathematics and Scientific Computing, Weierstrass Institute, Berlin, Germany

Abstract

This talk is based on the work of Alexander Linke & Christian Merdon. The talk discusses several physical regimes of the incompressible Navier–Stokes equations with respect to the role of the pressure and the role of the Helmholtz-Hodge projector in the Navier-Stokes momentum balance. It is emphasized that not the forces in the momentum balance themselves matter for the dynamics of the flow field, but their Helmholtz-Hodge projector. Since the Helmholtz-Hodge projector vanishes for arbitrary gradient fields, a semi norm and corresponding equivalence classes of forces play naturally a major role for the evolution of incompressible flows. Novel pressure-robust mixed finite element methods are designed for an appropriate discrete treatment of these equivalence classes of forces. On the contrary, classical, (only) inf-sup stable mixed methods do not care about the existence of such equivalence classes. In order to deliver accurate simulation results for the discrete velocities, they have to resort to expensive high order ansatz spaces, in order to reduce a corresponding consistency error of an appropriately defined discrete Helmholtz-Hodge projector. Last but not least, the talk will indicate suitable applications for efficient and accurate low-order pressure-robust mixed methods, like transient generalized Beltrami flows at high Reynolds numbers and electrohydrodynamics in micro- and nanofluidics. Reference: Nicolas R. Gauger, Alexander Linke, Philipp W. Schroeder: On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond. arXiv 1808.10711, 2018. Bio Dr. Linke's interests include: Numerical analysis for partial differential equations (PDEs), Incompressible Navier-Stokes equations & reaction-advection-diffusion equations, Finite Elements, Finite Volumes & Discontinuous Galerkin Methods. In his research, he tries to find robust discretizations for PDEs, which preserve fundamental qualitative properties of the continuous problem, like: Pressure-robust mixed methods for the incompressible Navier-Stokes equations (velocity error is independent of the pressure) and Discrete L∞-bounds, discrete maximum principles, and discrete positivity. He believes that the preservation of qualitative properties is of central importance for the robust discretization of multi physics problems. 2018 habilitation thesis: Towards Pressure-robust Discretizations for the Incompressible Navier-Stokes Equations. Free Universitity of Berlin, 2018 since 2016: tenured position as a scientific assistant at Weierstrass Institute, Berlin 2015 (summer semester): supply (associate) professorship for ‘Numerical analysis of PDEs’, TU Dresden 2008 Divergence-Free Mixed Finite Elements for the Incompressible Navier-Stokes Equation, PhD thesis, Friedrich-Alexander University of Erlangen-Nürnberg.