AMG Wave-Ray Solver for Helmholtz Eigenvalue Problem: Preliminary Results

Irene Livshits

Department of Mathematics, Univsersity of Central Arkansas,  Conway, AR 72035


Numerical solvers for indefinite Helmholtz equations have to deal with a range of erroneous components which have a slow convergence by standard relaxation procedures. Brandt and Livshits suggested wave-ray approach that allows to overcome such slowness by introducing ray representations and ray treatment of problematic components. This approach, however, does not have a straightforward extension for Helmholtz operators with variable coefficients, since it requires a knowledge of analytical solutions of homogeneous Helmholtz operator (principal components).

In this talk I will discuss the ways of modifying the wave-ray approach for solving eigenvalue problems for Helmholtz operators with variable coefficients. The presented algorithm employs Galerkin based algebraic multigrid, and modified wave-ray approach, which employs the best current approximation to the solution instead of principal components. The algorithm and the results of the numerical experiments will be discussed on the example of one-dimensional Helmholtz operator with periodic boundary conditions. Joint work with A.Brandt.