A Generalized Framework for Higher-Order Localized Orthogonal Decomposition Methods

TL;DR

This paper presents a unified framework for higher-order Localized Orthogonal Decomposition methods, enabling flexible constraints and improved localization strategies for complex PDEs, with comprehensive analysis and numerical validation.

Contribution

It introduces a generalized approach that unifies conforming and nonconforming constraints in multiscale methods, extending applicability to various PDEs.

Findings

Effective localization strategies for linear elliptic problems

Comparison between conforming and nonconforming constraints

Extensions to Helmholtz and Gross--Pitaevskii problems

Abstract

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.