When integrating unsteady problems with continuous finite element methods (or methods using a continuous representation of data), one faces the problem of inverting a mass matrix. In some cases, one has to recompute this mass matrix at each time steps, in some methods that are not directly formulated by standard variational principles, it is not clear how to write an invertible mass matrix. Hence, in this paper, we show how to avoid this problem for hyperbolic systems, and we also detail the conditions under which this is possible. Analysis and simulation support our conclusions, namely that it is possible to avoid inverting mass matrices without sacrifying the accuracy of the scheme. I will also present some results for systems (work in progress).

Professor Remi Abgrall is a well-known French scientist in the field of numerical analysis and methods for partial differential equations. He graduated from Ecole Normale Superieure de Saint Cloud (in Mathematics), and was awarded his PhD (in Mechanics) from University Pierre et Marie Curie (Paris) in 1988. He started his career at ONERA (France) as a research engineer, and then research scientist at Inria. He became a professor at the University of Bordeaux (France) from 1996 till 2013, then moved to Zurich in January 2013. He is known for his work on compressible flow simulation (ENO schemes on unstructured meshes, multimaterial and multiphase flow simulation, high-order schemes on unstructured meshes), approximation of Hamilton Jacobi equation, Uncertainty quantification, meshing of complex domain with curved meshes for high order schemes, etc).  He has honored with several awards, including invited speaker at the ICM in Seoul, 2014. He has been a plenary speaker at several international conferences, including ICCP7 (Beijing, 2010), WCCM IACM-ECCOMAS (Barcelona, 2014), XVI International Conference on Hyperbolic Problems: Theory, Numerics, Applications (Aachen, Germany, 2016).  He serves as a member of editorial boards of several journals (Mathematics of Computation, Journal of Scientific Computing, CiCP, Computers and Fluids, etc.) and is the current Editor in Chief of the Journal of Computational Physics.