Equation-based interpolation and incremental unknowns for solving the three-dimensional Helmholtz equation

Pascal Poullet, Amir Boag

Research output: Contribution to journalReview articlepeer-review


In an earlier paper (Poullet and Boag, 2007) [1], we developed an efficient incremental unknowns (IU) preconditioner for solving the two-dimensional (2D) Helmholtz problem in both high and low frequency (wavenumber) regimes. The multilevel preconditioning scheme involves separation of each grid into a coarser grid of the following level and a complementary grid on which the IUs are defined by interpolation. This approach is efficient as long as the mesh size of the coarsest grid is sufficiently small compared to the wavelength. In order to overcome this restriction, the authors introduced recently (in Poullet and Boag (2010) [2]) a modified IU method combining the conventional interpolation with the Helmholtz equation based interpolation (EBI). The EBI coefficients are derived numerically using a sufficiently large set of analytic solutions of the Helmholtz equation on a special hierarchy of stencils. The modified IUs using Helmholtz EBI are shown to provide improved preconditioning on the coarse scales where the conventional interpolation can not be employed. This study deals with the extension of this idea for solving the three-dimensional (3D) Helmholtz equation.

Original languageEnglish
Pages (from-to)1200-1208
Number of pages9
JournalApplied Mathematics and Computation
StatePublished - 1 Apr 2014


  • Helmholtz equation
  • Incremental Unknowns
  • Iterative methods
  • Multilevel methods
  • Preconditioning

All Science Journal Classification (ASJC) codes

  • Computational Mathematics
  • Applied Mathematics


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