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<p>We propose a spectral solver for the Poisson equation on a square domain, achieving optimal complexity through the ultraspherical spectral method and the alternating direction implicit (ADI) method. Compared with the state-of-the-art spectral solver for the Poisson equation by Fortunato and Townsend&#xa0;(IMA J Numer Anal 40(3):1994–2018 (2019)), our method not only eliminates the need for conversions between Chebyshev and Legendre bases but also is applicable to more general boundary conditions while maintaining spectral accuracy. We prove that, for solutions with sufficient smoothness, a fixed number of ADI iterations suffices to meet a specified tolerance, yielding an optimal complexity of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3011_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {O}(n^2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The solver can also be extended to other equations as long as they can be split into two one-dimensional operators with nearly real and disjoint spectra. Numerical experiments demonstrate that our algorithm can resolve solutions with millions of unknowns in under a minute, with significant speedups when leveraging low-rank approximations.</p>

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An Optimal Complexity Spectral Solver for the Poisson Equation

  • 期刊论文

  • 发表日期：

    2025年8月6日

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  • Ouyuan Qin

摘要

We propose a spectral solver for the Poisson equation on a square domain, achieving optimal complexity through the ultraspherical spectral method and the alternating direction implicit (ADI) method. Compared with the state-of-the-art spectral solver for the Poisson equation by Fortunato and Townsend (IMA J Numer Anal 40(3):1994–2018 (2019)), our method not only eliminates the need for conversions between Chebyshev and Legendre bases but also is applicable to more general boundary conditions while maintaining spectral accuracy. We prove that, for solutions with sufficient smoothness, a fixed number of ADI iterations suffices to meet a specified tolerance, yielding an optimal complexity of \(\mathcal {O}(n^2)\) O ( n 2 ) . The solver can also be extended to other equations as long as they can be split into two one-dimensional operators with nearly real and disjoint spectra. Numerical experiments demonstrate that our algorithm can resolve solutions with millions of unknowns in under a minute, with significant speedups when leveraging low-rank approximations.