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High-Resolution Solvers for 3D Helmholtz Scattering Problems Using PFFT and Eigenvector-Based Preconditioning
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DOI：10.1016/j.jcp.2026.114939.png)
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Abstract
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• This paper demonstrates the critical importance of consistent boundary condition representation in the preconditioner for high-order schemes with variable coefficients, a phenomenon also noted in [1]. Our key innovation is to include the low-order approximation of the local boundary condition operator B in the preconditioner. The resulting preconditioning matrix is inverted at each Krylov subspace step by a direct EigT or PFFT algorithm, extending methods proposed in [2], [3] for higher-order schemes with non-constant coefficients. The outer iterations are performed using the Generalized Minimal Residual (GMRES) method [4]. • To illustrate the efficiency of the new GMRES solvers with PFFTtype and EigT-type preconditioners, we conduct a comprehensive comparison with previously developed FFT-based methods. Our numerical experiments demonstrate that the proposed preconditioned GMRES solvers can outperform the standard GMRES-FFT combination [5], [6], [7] by close to an order of magnitude in the majority of test cases.
Keywords:
High-order schemes
Preconditioning
GMRES method
PFFT
EigT
Journal
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3.8
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1.5W
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7.4W