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An optimal complexity spectral solver for the Poisson equation

Ouyuan Qin

TL;DR

This work develops a unified, spectrally accurate Poisson solver on a square by marrying the ultraspherical spectral method with alternating direction implicit iterations. By deriving sharp spectrum bounds and employing Zolotarev-optimal shifts, the authors prove that a fixed number of ADI iterations suffices for sufficiently smooth solutions, yielding true $O(n^2)$ complexity. The method accommodates Neumann/Robin and separable coefficients, supports fourth-order and nonhomogeneous boundary conditions, and enables low-rank acceleration when the right-hand side is compressible. Numerical experiments show the solver handles millions of unknowns rapidly and outperforms competing approaches, with practical impact for high-accuracy PDE simulations and potential extensions to broader elliptic problems.

Abstract

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 \cite{for}, 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 $\mathcal{O}(n^2)$. 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.

An optimal complexity spectral solver for the Poisson equation

TL;DR

This work develops a unified, spectrally accurate Poisson solver on a square by marrying the ultraspherical spectral method with alternating direction implicit iterations. By deriving sharp spectrum bounds and employing Zolotarev-optimal shifts, the authors prove that a fixed number of ADI iterations suffices for sufficiently smooth solutions, yielding true complexity. The method accommodates Neumann/Robin and separable coefficients, supports fourth-order and nonhomogeneous boundary conditions, and enables low-rank acceleration when the right-hand side is compressible. Numerical experiments show the solver handles millions of unknowns rapidly and outperforms competing approaches, with practical impact for high-accuracy PDE simulations and potential extensions to broader elliptic problems.

Abstract

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 \cite{for}, 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 . 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.

Paper Structure

This paper contains 21 sections, 1 theorem, 67 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Ultraspherical spectral method
  3. ADI method
  4. Estimations for spectra
  5. Optimal complexity
  6. Implementation details
  7. Error check
  8. Warm restart
  9. Order of Shifts
  10. Low-Rank Forms
  11. Extensions
  12. Poisson equation with Neumann and Robin conditions
  13. Poisson equation with separable coefficients
  14. Fourth-order equation
  15. Nonhomogeneous boundary conditions
  16. ...and 6 more sections

Key Result

Theorem 1

With ass:smoothness and wbound, given $0< \epsilon < W_1 + W_2$, there is a positive number $k_{\epsilon}$ (independent of $n$) such that $\left\|E^{k_\epsilon}\right\| \leq \sqrt{3}\epsilon$.

Figures (5)

  • Figure 1: Left: the ratios of bounds in \ref{['useigbound2']} and extreme eigenvalues computed numerically for different finite truncations $n$. Right: relative true error and relative increment error during the ADI iterations when solving a Poisson equation with \ref{['alg:prototype']} and sufficiently large $n$.
  • Figure 2: Left: execution times for solving \ref{['ex1']} using new method with different tolerances $\epsilon$ and the FT method with the tightest tolerance. Right: differences between solutions of new method and that of the FT method.
  • Figure 3: Left: execution times for solving \ref{['ex2']} using new method with different tolerances and the B--S algorithm with new method and TO method. Right: relative errors of different methods.
  • Figure 4: Left: relative increment errors of ADI iterations for different initial iterations and different orders of application of shifts. Right: relative increment errors in the solution using warm restart and descending order of shifts.
  • Figure 5: Left: execution times against iterations for solving \ref{['fourth']} using fADI \ref{['fadiiter1']} and ADI \ref{['adiiter']}. Right: relative errors of different methods.

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof