Robert Scheichl: An Introduction to Multiscale Methods and Localised Model Reduction (Part III) скачать в хорошем качестве

Robert Scheichl: An Introduction to Multiscale Methods and Localised Model Reduction (Part III)

This series of lectures explores the development of multiscale methods and localised model reduction for the solution of PDEs that involve coefficients that vary across multiple scales. The variation may be inherent in the problem, e.g., when modelling composite materials, or it may arise in the context of uncertainty quantification. In both cases, this leads to high-dimensional parametric PDEs with multiscale behaviour that does not separate and is not easily amenable to homogenisation techniques. Due to the lack of scale separation, it is crucial that the local multiscale behaviour is captured well enough in the computational approach, but explicit discretisation with classical (polynomial) Finite Element Methods (FEM) is prohibitive. Using the framework of Generalised Finite Element Methods (GFEM), I will show how local fine scale information can be incorporated in the approximation space and how this can then be embedded in an efficient localised model reduction framework to develop surrogates for high-dimensional parameter spaces. This will encompass methods, such as the Multiscale FEM, the Generalised Multiscale FEM, Localised Orthogonal Decomposition (LOD), and the Multiscale-Spectral Generalised FEM.