A Generalized Framework for Higher-Order Localized Orthogonal Decomposition Methods

We introduce a generalized framework for studying higher-order versions of the multiscale method known as Localized Orthogonal Decomposition. Through a suitable reformulation, we are able to accommodate both conforming and nonconforming constraints in the construction process. In particular, we offer a new perspective on localization strategies. We fully analyze the strategy for linear elliptic problems and discuss extensions to the Helmholtz equation and the Gross–Pitaevskii eigenvalue problem. Numerical examples are presented that particularly provide valuable comparisons between conforming and nonconforming constraints.