YoungJoon Hong, Indiana University

Title: Analysis and computations of convection dominated flows in the presence of a boundary: boundary layers, atmospheric equations. 
Abstract: In this talk, I will present convergence results of singularly perturbed problems in the sense of PDEs, which is related to the vanishing viscosity limit. I also provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the socalled boundary layer elements which absorb the boundary layer singularities. Using a P1 classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasiuniform mesh. In the second part of my talk, I will present a finite volume scheme to solve the two dimensional inviscid primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically relevant boundary conditions. In that respect, a version of a projection method is introduced to enforce the compatibility condition on the horizontal velocity field, which comes from the boundary conditions.
