Robert Scheichl (University of Heidelberg)


Title: Multiscale-Spectral Generalized Finite Elements: Efficient Localized Model Order Reduction
Abstract: In this talk, I will present an efficient generalized finite element method with optimal (multiscale spectral) local approximation spaces (MS-GFEM) for PDEs with heterogeneous coefficients. In practice, the local approximation spaces are constructed from local eigenproblems solved on some sufficiently fine finite element mesh with mesh size h. In this work, we will provide rigorous error estimates for the fully discrete method. The error bound of the discrete MS-GFEM approximation is proved to converge as h tends to 0 towards that of the continuous MS-GFEM approximation, which was shown to decay nearly exponentially in previous works. Moreover, even for fixed mesh size h, a nearly exponential rate of convergence of the local approximation errors with respect to the dimension of the local spaces is established. The method can also be used as an effective preconditioner in an iterative solver with a ’tuneable’ convergence rate. On the practical side, an efficient and accurate method for solving the discrete eigenvalue problems is proposed by incorporating the discrete-harmonic constraint directly into the eigenproblem via a Lagrange multiplier approach. Numerical experiments that confirm the theoretical results and demonstrate the effectiveness of the method are presented for a second-order elliptic problem, for a large-scale problem of linear elasticity in composite aerospace applications and for a high-frequency heterogeneous Helmholtz problem. Even in this last example, a quasi-optimal and nearly exponential (wavenumber-explicit) global convergence of the method can be theoretically proved, provided the size of the subdomains is O(1/k) (where k is the wavenumber).

(Host: Markus Bachmayr)