Axel Målqvist - Localization of Multiscale Problems

Explore a comprehensive lecture on the localization of multiscale problems, delivered by Axel Målqvist as part of the Hausdorff Trimester Program. Delve into the Local Orthogonal Decomposition technique for solving partial differential equations with multiscale data, with a focus on convergence in high contrast scenarios. Examine various applications, including time-dependent problems where the computed basis can be reused in each time step. Cover topics such as elliptic model problems, multiscale decomposition, orthogonalization, ideal multiscale representation, fine scale discretization, a priori error analysis, 3D implementation in Python, high contrast data Poisson equation, Scott-Zhang type interpolation, geometry-dependent interpolation, numerical examples for eigenvalues, parabolic equations, and a numerical experiment on the heat equation. Gain valuable insights into this complex field of mathematics through this 58-minute presentation from the Hausdorff Center for Mathematics.

Intro
Outline
Elliptic model problem
Multiscale decomposition
Orthogonalization
Ideal multiscale representation
Fine scale discretization
A priori error analysis
3D implementation in python
High contrast data Poisson equation
Scott-Zhang type interpolation Nodal variables
Geometry dependent interpolation Selection of cry
Numerical example: eigenvalues
Parabolic equations
Numerical experiment: The heat equation
Conclusion