High-Resolution Solvers for 3D Helmholtz Scattering Problems Using PFFT and Eigenvector-Based Preconditioning
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914314281025536 |
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| author | Gryazin, Yury Gonzales, Ron Li, Xiaoye Sherry |
| author_facet | Gryazin, Yury Gonzales, Ron Li, Xiaoye Sherry |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07711 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | High-Resolution Solvers for 3D Helmholtz Scattering Problems Using PFFT and Eigenvector-Based Preconditioning Gryazin, Yury Gonzales, Ron Li, Xiaoye Sherry Numerical Analysis 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. |
| title | High-Resolution Solvers for 3D Helmholtz Scattering Problems Using PFFT and Eigenvector-Based Preconditioning |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2602.07711 |