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

Explore the development of multiscale methods and localised model reduction for solving partial differential equations (PDEs) with coefficients varying across multiple scales in this comprehensive lecture. Learn how to address high-dimensional parametric PDEs with multiscale behavior that doesn't separate and isn't easily amenable to homogenization techniques, particularly when modeling composite materials or in uncertainty quantification contexts. Discover why explicit discretization with classical polynomial Finite Element Methods (FEM) becomes prohibitive due to lack of scale separation, and understand the crucial importance of capturing local multiscale behavior in computational approaches. Master the framework of Generalised Finite Element Methods (GFEM) to incorporate local fine scale information into approximation spaces and embed this within efficient localised model reduction frameworks for developing surrogates in high-dimensional parameter spaces. Examine key methodologies including Multiscale FEM, Generalised Multiscale FEM, Localised Orthogonal Decomposition (LOD), and Multiscale-Spectral Generalised FEM, gaining practical insights into how these advanced techniques solve complex multiscale problems where traditional methods fall short.