Error Analysis of a Variational Multiscale Stabilization for Convection-Dominated Diffusion Equations in 2D

Explore a 35-minute lecture on error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in 2D. Delve into the formulation of a stabilized quasi-optimal Petrov-Galerkin method based on the variational multiscale approach. Examine the exponential decay of fine-scale correctors and their localization to patch problems dependent on velocity field direction and singular perturbation parameter. Discover how this stabilization ensures stability and quasi-optimal convergence rates for arbitrary mesh Péclet numbers on coarse meshes. Learn about the method's performance, impact of interpolation operators, element patches, and localized multiscale bases through numerical simulations and comprehensive analysis.

Overview
Background
Motivation
Brief literature on multiscale methods
Upper bounds in V and Vb
Spectral gap under certain assumption
Conformal Galerkin approximation
Performance of the Galerkin approximation
State of art methods
Nodal interpolation based VMS
An illustration of multiscale test bases
Impact of the choice of interpolation operators
Element patches
Element correctors and localized multiscale bases
An illustration of the localized multiscale test bases
Localized VMS for the model problem
Numerical simulation
Conclusion 2