Title: Involution and Invariant-Domain Preserving Approximation of the Ideal Magneto-Hydrodynamics Equations

Abstract: I will discuss recent results regarding the approximation of Maxwell's equations written in first-order form. I will show in particular that the discontinuous Galerkin approximation is spectrally correct. This result is based on the observation that the discontinuous Galerkin approximation preserves a fundamental involution property which is known as Gauss's law in the literature. A similar result is conjectured to hold true for stabilized continuous finite elements. I will also present a numerical technique for solving the ideal MHD equations that is invariant-domain preserving and involution preserving. The key novelty is that no divergence cleaning is needed.  The method will be numerically illustrated using continuous finite elements. The first part of this project is a joint work with Alexandre Ern (ENPC, Paris).  The second part of this project is a joint work with Tuan Anh Dao (Uppsala University) and Ignacio Tomas (Texas Tech University).

Bio: Dr. Jean-Luc Guermond joined Texas A&M’s Department of Mathematics in 2004 and has been the Richard E. Ewing ExxonMobil Chair since 2012. Dr. Guermond’s research interests include numerical analysis and partial differential equations. He received his Ph.D. from the University of Paris VI before becoming a research associate in the Department of Ocean Engineering at MIT and the Computer Science Laboratory for Mechanics and Engineering Sciences in Orsay, France, amongst others. Dr. Guermond has served as International Chair for Inria and received grants from the Air Force Office of Scientific Research, Army Research Office, Department of Energy, and National Science Foundation. His is a Fellow of both the American Mathematical Society and the Society of Industrial and Applied Mathematics. He is a managing editor for various journals.