Multiscale Finite Element Methods for Advection and Reaction-Diffusion Problems

Explore multiscale finite element methods (MsFEM) for solving partial differential equations with highly oscillatory coefficients on coarse meshes. Dive into the first part of the lecture, focusing on multiscale advection-diffusion problems in convection-dominated regimes, and learn about different approaches to define MsFEM basis functions and combine them with stabilization techniques. Discover how methods using bubble functions and Crouzeix-Raviart type boundary conditions prove highly effective. In the second part, examine reaction-diffusion equations with oscillating coefficients, framed as eigenvalue equations. Gain insights into the application of theoretical homogenization results in periodic frameworks to guide the definition of appropriate MsFEM basis functions. Understand efficient problem-solving techniques presented by the speaker, drawing from collaborative work with Rutger Biezemans, Claude Le Bris, Alberic Lefort, and Alexei Lozinski.