A generalized framework for higher-order Localized Orthogonal Decomposition methods

We revisit the higher-order Localized Orthogonal Decomposition variant by Maier [ SIAM J. Numer. Anal. 59 (2021) 1067–1089] based on nonconforming constraints (discontinuous finite element spaces) and introduce a new variant based on conforming constraints (continuous finite elements), putting both approaches in a general unified framework. We propose a new localization strategy that is suitable for both approaches and offers a new perspective on the localization of LOD in general. We fully analyze the strategy for linear scalar elliptic problems and discuss extensions to the Helmholtz equation and the Gross–Pitaevskii eigenvalue problem. Numerical examples are presented that provide valuable comparisons between conforming and nonconforming constraints.