Open Access

Hannah Mohr

• Many physical phenomena in science and engineering, such as groundwater flow or the behavior of composite materials, are governed by partial differential equations with highly heterogeneous coefficients acting on multiple spatial scales. Direct numerical simulation of such multiscale problems is computationally demanding, as classical methods like the finite element method require resolving the finest scales. Although analytical homogenization provides effective macroscopic models under strong assumptions such as periodicity or scale separation, these assumptions are often violated in realistic applications. This thesis addresses the numerical homogenization of elliptic diffusion problems with highly heterogeneous deterministic and stochastic coefficients. We focus on the Super-Localized Orthogonal Decomposition (SLOD) method, which enables accurate coarse-scale approximations without relying on restrictive structural assumptions. SLOD extends the classical Localized OrthogonalMany physical phenomena in science and engineering, such as groundwater flow or the behavior of composite materials, are governed by partial differential equations with highly heterogeneous coefficients acting on multiple spatial scales. Direct numerical simulation of such multiscale problems is computationally demanding, as classical methods like the finite element method require resolving the finest scales. Although analytical homogenization provides effective macroscopic models under strong assumptions such as periodicity or scale separation, these assumptions are often violated in realistic applications. This thesis addresses the numerical homogenization of elliptic diffusion problems with highly heterogeneous deterministic and stochastic coefficients. We focus on the Super-Localized Orthogonal Decomposition (SLOD) method, which enables accurate coarse-scale approximations without relying on restrictive structural assumptions. SLOD extends the classical Localized Orthogonal Decomposition method by constructing basis functions with super-exponential decay through carefully designed local source terms. We analyze its approximation properties and propose stabilization strategies to ensure numerical robustness. Based on SLOD, we introduce a multilevel extension called Hierarchical SLOD (HSLOD). This approach constructs quasi-orthogonal hierarchical basis functions, allowing for a multiresolution decomposition of the solution space. The hierarchical structure improves the conditioning of the resulting linear systems, supports efficient parallel solvers, and enables incremental refinement by adding further discretization levels. In the stochastic setting, we extend SLOD and HSLOD to a collocation-type framework for numerical stochastic homogenization. The proposed methods efficiently compute expected solutions of PDEs with random coefficients by combining the super-exponential localization of the basis functions with the simplicity of collocation approaches, which avoid assembling global stiffness matrices. Rigorous error estimates are derived using results from quantitative stochastic homogenization theory, and the theoretical findings are validated through extensive numerical experiments.…