Holger Heumann, visiting professor at Rutgers University. will address "Lagrangian and Eulerian Methods for Generalized Advection-Diffusion" on April 19, noon to 2 p.m., in Donovan Hall Rm. G152.

Abstract:"We study stationary and transient advection diffusion equations for differential forms and their Lagrangian and Eulerian discretization by means of discrete differential forms. The discretizations with discrete differential forms provide models both for the classical scalar advection diffusion equation, as well as for the transport of magnetic fields in moving conducting fluids (magnetohydrodynamics). The challenge is robustness and optimal a priori error estimates in the case of vanishing diffusion. Exterior calculus involving the Lie derivative will be harnessed to achieve streamlined formulations and sharper a priori convergence  estimates."

Heumann received a Ph.D. in mathematics from Swiss Federal Institute of Technology in Zurich (ETH) in 2011, where he developed and implemented efficient multi-grid methods or stable mixed finite element methods for magnetohydrodynamics. For this he used exterior calculus of differential forms, a coordinate-free approach to multivariable calculus invented by E. Cartan a century ago, which offers a unified approach to integration over curves, surfaces and higher dimensional manifolds, and has recently received much attention in the construction of structure preserving numerical methods.

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At the LRC Fusion in Nice, France, Heumann contributed to the development of software for plasma and nuclear fusion simulations. He has worked with Ralf Hiptmair, Jacques Blum and Michael Vogelius; he was an Oberwolfach Leibniz graduate student and was awarded a fellowship by the Chinese University of Hong Kong. He was also the lead developer of LehrFEM, a MATLAB finite element library for teaching numerical mathematics.