Seminario de Matemática Aplicada: Compatible finite element discretization of the Lie advection-diffusion problem with application to magnetohydrodynamics

Enrico Zampa. Department of Mathematics, University of Trento The Lie advection-diffusion problem arises from many physical models, such as magnetohydrodynamics, two-phase flows, the Godunov-Peshkov-Romenskii model, etc. Devising a numerical discretization of this problem which is at the same time stable in absence of diffusion, high-order accurate and involution-preserving is far from trivial. In this presentation, I will propose two different approaches based on compatible finite elements. The first builds upon the interpolation-contraction method proposed by Hiptmair and Pagliantini, based on an approximate Rusanov-like multi-dimensional Riemann solver. The second approach can be seen as a generalization of the classical finite element stabilizations (SUPG, CIP…) to compatible finite elements. To mitigate the Gibbs phenomenon, both methods can be enriched with an a posteriori MOOD-style artificial resistivity guided by a discrete maximum principle. Finally, we show how the aforementioned methods can be integrated with a staggered semi-implicit hybrid finite volume/finite element solver for the incompressible Navier-Stokes equations, to obtain a novel scheme for incompressible magnetohydrodynamics.