On the indefinite Helmholtz equation: complex stretched absorbing boundary layers, iterative analysis, and preconditioning

TL;DR

This paper introduces a new preconditioning approach for Helmholtz problems with complex stretched absorbing layers, using a multigrid cycle, and compares its performance to existing methods through numerical analysis.

Contribution

It proposes a novel preconditioner for Helmholtz equations involving complex coordinate stretching, invertible via multigrid, offering an alternative to the complex shifted Laplacian.

Findings

Preconditioner is invertible with multigrid cycle.

Performance is comparable to complex shifted Laplacian.

Numerical analysis based on eigenvalues supports effectiveness.

Abstract

This paper studies and analyzes a preconditioned Krylov solver for Helmholtz problems that are formulated with absorbing boundary layers based on complex coordinate stretching. The preconditioner problem is a Helmholtz problem where not only the coordinates in the absorbing layer have an imaginary part, but also the coordinates in the interior region. This results into a preconditioner problem that is invertible with a multigrid cycle. We give a numerical analysis based on the eigenvalues and evaluate the performance with several numerical experiments. The method is an alternative to the complex shifted Laplacian and it gives a comparable performance for the studied model problems.