A reciprocal preconditioner for structured matrices arising from elliptic problems with jumping coefficients

Sergey Dolgov, Boris N. Khoromskij, Ivan Oseledets, Eugene Tyrtyshnikov

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Abstract

We present a new explicit preconditioner for a class of structured matrices Γ(a) defined by a function a. These matrices may arise from elliptic problems by a standard discretization scheme admitting jumps in the diffusion coefficient. When the grid is fixed, the matrix is determined by the diffusion coefficient a. Our preconditioner P is of the form P=Γ- 1(1)Γ(1/a)Γ- 1(1) and can be considered as an approximation to the inverse matrix to Γ(a). We prove that P and Γ- 1(a) are spectrally equivalent. However, the most interesting observation is that Γ(a)P has a cluster at unity. In the one-dimensional case this matrix is found to be equal to the identity plus a matrix of rank one. In more dimensions, a rigorous proof is given in the two-dimensional stratified case. Moreover, in a stratified case with M constant values for the coefficient a, a hypothesis is proposed that a wider set of M+1 points including unity is a proper cluster. In such cases the number of iterations does not depend dramatically on jumps of the diffusion coefficient. In more general cases, fast convergence is still demonstrated by many numerical experiments.

Original languageEnglish
Pages (from-to)2980-3007
Number of pages28
JournalLinear Algebra and its Applications
Volume436
Issue number9
Early online date13 Oct 2011
DOIs
Publication statusPublished - 1 May 2012

Bibliographical note

Funding Information:
Corresponding author. E-mail addresses: sergey.v.dolgov@gmail.com (S. Dolgov), bokh@mis.mpg.de (B.N. Khoromskij), ivan.oseledets@gmail.com (I. Oseledets), tee@bach.inm.ras.ru (E. Tyrtyshnikov). 1 Supported by the RFBR grants 11-01-12137, 11-01-00549-a, 09-01-91332 (joint with DFG), the Government Contracts Π940, Π1112, 14.740.11.0345 and Priority Research Grant of the Presidium and of the Department of Mathematical Sciences of the Russian Academy of Sciences. 2 During this work the author was also a visiting professor at the University of Siedlce (Poland) and University of Chester (UK).

Keywords

  • Elliptic operators
  • Finite differences
  • Finite elements
  • Iterative methods
  • Multi-dimensional matrices
  • Numerical methods
  • Poisson equation
  • Preconditioners
  • Structured matrices

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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