Numerical homogenization: multi-resolution and super-localization approaches

Multi-scale problems arise in many scientific and engineering applications, where the effective behavior of a system is determined by the interaction of effects at multiple scales. To accurately simulate such problems without globally resolving all microscopic features, numerical homogenization techniques have been developed. One such technique is the Localized Orthogonal Decomposition (LOD). It provides reliable approximations at coarse discretization levels using problem-adapted basis functions obtained by solving local sub-scale correction problems. This allows the treatment of problems with heterogeneous coefficients without structural assumptions such as periodicity or scale separation. This thesis presents recent achievements in the field of LOD-based numerical homogenization. As a starting point, we introduce a variant of the LOD and provide a rigorous error analysis. This LOD variant is then extended to the multi-resolution setting using the Helmholtz problem as a model problem. The multi-resolution approach allows to improve the accuracy of an existing LOD approximation by adding more discretization levels. All discretization levels are decoupled, resulting in a block-diagonal coarse system matrix. We provide a wavenumber-explicit error analysis that shows convergence under mild assumptions. The fast numerical solution of the block-diagonal coarse system matrix with a standard iterative solver is demonstrated. We further present a novel LOD-based numerical homogenization method named Super-Localized Orthogonal Decomposition (SLOD). The method identifies basis functions that are significantly more local than those of the LOD, resulting in reduced computational cost for the basis computation and improved sparsity of the coarse system matrix. We provide a rigorous error analysis in which the stability of the basis is quantified a posteriori. However, for challenging problems, basis stability issues may arise degrading the approximation quality of the SLOD. To overcome these issues, we combine the SLOD with a partition of unity approach. The resulting method is conceptually simple and easy to implement. Higher order versions of this method, which achieve higher order convergence rates using only the regularity of the source term, are derived. Finally, a local reduced basis (RB) technique is introduced to address the challenges of parameter-dependent multi-scale problems. This method integrates a RB approach into the SLOD framework, enabling an efficient generation of reliable coarse-scale models of the problem. Due to the unique localization properties of the SLOD, the RB snapshot computation can be performed on particularly small patches, reducing the offline and online complexity of the method. All theoretical results of this thesis are supported by numerical experiments

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Numerical Simulation of Multi-Phase Multi-Component Reactive Flow in Porous Media

In this thesis we develop a multi-component multi-phase reactive transport simulator to facilitate the investigation of a large variety of phenomena in porous media including component transport, diffusion, microbiological growth and decay, cell attachment and detachment and phase exchange. The coupled problem is solved using operator splitting approach. This approach enables us to use higher-order schemes and reduce numerical diffusion, which can result in an overestimation of phase exchange and reaction processes. Furthermore, this approach allows a flexible adaptation of the solution strategy to the concrete problem. We conduct an in-depth comparison of the fully-coupled and splitting approaches in order to derive criteria for the most efficient scheme depending on the relative importance of advection, diffusion and reaction