A reduced basis localized orthogonal decomposition

TL;DR

This paper introduces a novel method combining Reduced Basis and Localized Orthogonal Decomposition techniques to efficiently solve parametrized multiscale elliptic problems, reducing computational costs for multiple parameter queries.

Contribution

The paper presents a new RB-LOD approach that precomputes local basis functions offline for representative parameters, enabling fast online solutions for various parameters.

Findings

The method achieves significant computational savings in multiscale problems.

It provides sparse system matrices due to local support basis functions.

Applicable to both linear and nonlinear multiscale problems.

Abstract

In this work we combine the framework of the Reduced Basis method (RB) with the framework of the Localized Orthogonal Decomposition (LOD) in order to solve parametrized elliptic multiscale problems. The idea of the LOD is to split a high dimensional Finite Element space into a low dimensional space with comparably good approximation properties and a remainder space with negligible information. The low dimensional space is spanned by locally supported basis functions associated with the node of a coarse mesh obtained by solving decoupled local problems. However, for parameter dependent multiscale problems, the local basis has to be computed repeatedly for each choice of the parameter. To overcome this issue, we propose an RB approach to compute in an "offline" stage LOD for suitable representative parameters. The online solution of the multiscale problems can then be obtained in a coarse space (thanks to the LOD decomposition) and for an arbitrary value of the parameters (thanks to a suitable "interpolation" of the selected RB). The online RB-LOD has a basis with local support and leads to sparse systems. Applications of the strategy to both linear and nonlinear problems are given.