Fast homogenization algorithm based on asymptotic theory and multiscale schemes

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• 作者：Maria Morandi Cecchi and Massimo Fornasier
• 关键词：homogenization ; multiresolution analysis ; differential equations ; asymptotic analysis
• 刊名：Numerical Algorithms
• 出版年：2005
• 出版时间：October 2005
• 年：2005
• 卷：40
• 期：2
• 页码：171-186
• 全文大小：399 KB
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Algorithms
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9265

文摘

A time-frequency interpretation of the classical asymptotic theory of homogenization for elliptic PDE with periodic coefficients is presented and the relations with known multilevel/multiscale numerical schemes are investigated. We formulate a new fast iterative algorithm for the approximation of homogenized solutions based on the combination of these two apparently different approaches. The asymptotic homogenization process is interpreted as a migration to infinity of the frequencies related to microscale contributions and the discovering of those related to the homogenized solution. At different scale/frequency of the periodic coefficients of the operator, band-pass filters select only the contributions of the homogenized solution which is then composed as the limit of an iterative procedure. This novel method can be interpreted in case of finite difference discretizations as a generalized nonstationary subdivision scheme and its convergence and stability are discussed. In particular, stable compositions of the homogenized solution are investigated in relation with the contracting behavior of specific operators generated by reduction processes and Schur's complements of suitable matrices produced by discretizations via wavelets and multiscale bases.