A reduced basis localized orthogonal decomposition

JOURNAL OF COMPUTATIONAL PHYSICS

卷:295

Article

Abdulle, Assyr¹  Henning, Patrick² 

[1] Ecole Polytech Fed Lausanne, ANMC, Sect Math, CH-1015 Lausanne, Switzerland

[2] Univ Munster, Inst Numer & Angew Math, D-48149 Munster, Germany

关键词: Finite element;    Reduced basis;    Parameter dependent PDE;    Numerical homogenization;    Multiscale method;   

DOI  :  10.1016/j.jcp.2015.04.016

来源: Elsevier

【 摘 要 】

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. (C) 2015 Elsevier Inc. Allrights reserved.

【 授权许可】

Free