An Introduction to Multiscale Methods and Localised Model Reduction - Part III

Explore advanced multiscale methods and localised model reduction techniques for solving partial differential equations with coefficients varying across multiple scales in this third part of a comprehensive lecture series. Delve into the challenges of modeling composite materials and uncertainty quantification scenarios where traditional homogenisation techniques fall short due to lack of scale separation. Learn how Generalised Finite Element Methods (GFEM) framework incorporates local fine scale information into approximation spaces, moving beyond the limitations of classical polynomial Finite Element Methods. Discover efficient localised model reduction frameworks for developing surrogates in high-dimensional parameter spaces, covering essential methods including Multiscale FEM, Generalised Multiscale FEM, Localised Orthogonal Decomposition (LOD), and Multiscale-Spectral Generalised FEM. Understand how these computational approaches capture crucial local multiscale behavior while maintaining computational efficiency for complex parametric PDEs with inherent multiscale characteristics.