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Preconditioners based on windowed Fourier frames applied to elliptic partial differential equations

  • Samir K. BhowmikEmail author
  • Christiaan C. Stolk
Open Access
Article

Abstract

We investigate the application of windowed Fourier frames to the numerical solution of partial differential equations, focussing on elliptic equations. The action of a partial differential operator (PDO) on a windowed plane wave is close to a multiplication, where the multiplication factor is given by the symbol of the PDO evaluated at the wave number and central position of the windowed plane wave. This can be exploited in a preconditioning method for use in iterative inversion. For domains with periodic boundary conditions we find that the condition number with the preconditioning becomes bounded and the iteration converges well. For problems with a Dirichlet boundary condition, some large and small singular values remain. However the iterative inversion still appears to converge well.

Keywords

Windowed Fourier frame Symbol Elliptic PDE Preconditioner 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of DhakaDhakaBangladesh
  2. 2.KdV Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

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