Efficient implementation of the Localized Orthogonal Decomposition method

TL;DR

This paper introduces efficient algorithms for implementing the Localized Orthogonal Decomposition (LOD) method, enabling multiscale PDE simulations within standard finite element frameworks for various problem types.

Contribution

It provides practical algorithms for implementing LOD efficiently in finite element software, applicable to elliptic and eigenvalue problems with multiscale features.

Findings

Efficient algorithms for LOD implementation are developed.

The method is adaptable to different PDE types.

Implementation within standard finite element frameworks is demonstrated.

Abstract

In this paper we present algorithms for an efficient implementation of the Localized Orthogonal Decomposition method (LOD). The LOD is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales. We show how the method can be implemented in a fairly standard Finite Element framework and discuss its realization for different types of problems, such as linear elliptic problems with rough coefficients and linear eigenvalue problems.