腾讯会议ID：125 424 154

https://meeting.tencent.com/s/PFOoJGUPECca

Convection-diffusion problems, especially the convection dominated ones, are known to have many important applications and numerical challenges. In this talk, we present a robust discretization and solver developed for convection-dominated PDEs discretized on unstructured simplicial grids. The proposed methods can be applied to any one of the following operators: gradient, curl, and divergence. The derivation of the lowest order scheme makes use of some intrinsic properties of differential forms and in particular some crucial identities from differential geometry. We further give a systematic way for deriving high order schemes, by considering the properties of quasi-polynomial spaces defined as (exponentially) weighted spaces with polynomial coefficients. The analysis can be generalized to discrete differential forms of arbitrary order in any spatial dimension and any quasi-polynomial Hilbert complex of the first kind (Nédélec–Raviart–Thomas) or second kind (Nédélec–Brezzi–Douglas–Marini). Both theoretical analysis and numerical experiments show that the new upwinding finite element schemes provide an accurate and robust discretization and a fast solver in many applications and in particular for simulation of magnetohydrodynamics systems when the magnetic Reynolds number Rm is large.