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

SUMMARY 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.