High-Resolution Solvers for 3D Helmholtz Scattering Problems Using PFFT and Eigenvector-Based Preconditioning

TL;DR

This paper develops a high-resolution iterative solver for 3D Helmholtz problems that combines advanced finite-difference schemes with innovative eigenvector and PFFT-based preconditioners, improving efficiency and accuracy.

Contribution

It introduces two novel preconditioners based on eigenvector transformation and PFFT, tailored for high-resolution schemes solving 3D Helmholtz equations with absorbing boundaries.

Findings

Significant reduction in pollution error with high-order schemes.

Preconditioners improve convergence rates for large, ill-conditioned systems.

Numerical experiments confirm efficiency for realistic problem sizes.

Abstract

This paper presents an efficient Krylov subspace iterative solver for the three-dimensional (3D) Helmholtz equation with non-constant coefficients and absorbing boundary conditions, combining high-resolution compact schemes with low-order preconditioners. To mitigate numerical dispersion and reduce pollution error, we employ fourth- and sixth-order compact finite-difference schemes, thereby significantly softening the strict points-per-wavelength requirement. The resulting large, ill-conditioned linear systems are solved using a preconditioned GMRES method. The key innovation lies in the construction of the preconditioner: we introduce two highly efficient direct solvers - one based on a low-dimensional eigenvector transformation (EigT) and another on a partial Fast Fourier Transform (PFFT) algorithm - both derived from a lower-order approximation of the original problem that incorporates the absorbing boundary conditions. The motivation and efficacy of this lower-order preconditioning strategy for high-resolution schemes are analyzed through model problems, providing insight into the convergence rate. The theoretical analysis is validated by a comprehensive set of numerical experiments, demonstrating the method's performance for realistic problem sizes and parameters.