A scalable multigrid method for solving indefinite Helmholtz equations with constant wave numbers


  • Ira Livshits

    Corresponding author
    1. Department of Mathematical Sciences, Ball State University, Muncie, IN, USA
    • Correspondence to: Ira Livshits, Department of Mathematical Sciences, Ball State University, Muncie, IN 47304, USA.

      E-mail: ilivshits@bsu.edu

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Multigrid techniques are applied to two-dimensional indefinite Helmholtz equations, with a standard geometric approach aided by a special treatment of oscillatory near-kernel components of the Helmholtz operator. The algorithm is a modification of the wave-ray method by Brandt and Livshits, 1997. Its biggest difference from the original is a more traditional problem formulation: the Helmholtz equations with standard, in applications, Sommerfeld boundary conditions. The new slimmed wave-ray solver is less technical, and it is easier to implement. The algorithm is tested for a combination of problems and discretization parameters, and in all experiments with constant wave numbers, it shows stable convergence and scalability, all at computational costs comparable to the ones of the V (1,1) multigrid cycle. While not the main focus of the paper, Helmholtz equations with variable coefficients are considered, and some preliminaries are presented, followed by a discussion of the limited applicability of the method for such problems and the ways to expand it. Also briefly outlined is the potential and plans for a parallel implementation. Copyright © 2014 John Wiley & Sons, Ltd.