Freese, Jan PhilipJan PhilipFreeseHauck, MoritzMoritzHauckKeil, TimTimKeilPeterseim, DanielDanielPeterseim2024-01-032024-01-032024-02Numerische Mathematik 156 (1): 205-235 (2024-02)https://hdl.handle.net/11420/44872This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method’s basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method’s applicability for challenging high-contrast channeled coefficients.en0029-599X20241205235Springerhttps://creativecommons.org/licenses/by/4.0/65N1265N30MathematicsJournal Article10.15480/882.900910.1007/s00211-023-01386-410.15480/882.9009Journal Article