Christian Alber will talk about 'Multiscale Spectral Generalised Finite Elements: Preconditioning and Model Reduction Beyond SPD'
Multiscale PDEs with heterogeneous, highly oscillatory coefficients can be challenging for standard numerical methods. This talk presents a framework for the design and analysis of the Multiscale Spectral Generalised Finite Element Method (MS-GFEM), a partition of unity method using optimal local approximation spaces constructed from eigenproblems. These spaces are closely related to GenEO coarse spaces in domain decomposition. Under suitable assumptions, we establish nearly exponential local approximation rates and describe their dependence on problem parameters. We also discuss how the resulting coarse spaces can be used in two-level domain decomposition preconditioners. Applications to Helmholtz and Stokes problems illustrate the approach beyond the symmetric positive definite setting, with numerical experiments complementing the analysis.