This paper introduces a new sweeping preconditioner for the iterative solution of the variable coefficient Helmholtz equation in two and three dimensions. The algorithms follow the general structure of constructing an approximate LDLt factorization by eliminating the unknowns layer by layer starting from an absorbing layer or boundary condition. The central idea of this paper is to approximate the Schur complement matrices of the factorization using moving perfectly matched layers (PMLs) introduced in the interior of the domain. Applying each Schur complement matrix is equivalent to solving a quasi-1D problem with a banded LU factorization in the 2D case and to solving a quasi-2D problem with a multifrontal method in the 3D case. The resulting preconditioner has linear application cost, and the preconditioned iterative solver converges in a number of iterations that is essentially independent of the number of unknowns or the frequency. Numerical results are presented in both two and three dimensions to demonstrate the efficiency of this new preconditioner.

MSC codes

  1. 65F08
  2. 65N22
  3. 65N80


  1. Helmholtz equation
  2. perfectly matched layers
  3. high frequency waves
  4. preconditioners
  5. LDLt factorization
  6. Green’s function
  7. multifrontal methods
  8. optimal ordering

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Information & Authors


Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 686 - 710
ISSN (online): 1540-3467


Submitted: 6 August 2010
Accepted: 7 April 2011
Published online: 28 June 2011

MSC codes

  1. 65F08
  2. 65N22
  3. 65N80


  1. Helmholtz equation
  2. perfectly matched layers
  3. high frequency waves
  4. preconditioners
  5. LDLt factorization
  6. Green’s function
  7. multifrontal methods
  8. optimal ordering



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