On the Pre-Asymptotic Stability and Inverse Structure of Extended-Domain Spectral Methods

TL;DR

The paper analyzes extended-domain spectral collocation methods, showing that Fourier-extension frames induce asymptotic ill-conditioning for both Poisson and convection-diffusion operators when the system is square. It unveils a pre-asymptotic stability mechanism for convection-diffusion, rooted in a structural dichotomy of the discrete Green's function: the CD inverse is numerically quasi-sparse with exponential decay, unlike Poisson’s dense inverse. This dichotomy is demonstrated via a decay analysis of cardinal functions, a Jaffard-type inverse decay argument, and a discrete maximum-principle framework, with extensions to higher dimensions. Numerical validations in 1D and 2D confirm the theoretical predictions, showing exponential Lebesgue-constant growth but markedly better pre-asymptotic conditioning for CD, guiding stabilization and parameter choices in practice.

Abstract

The extended-domain method is a strategy for applying spectral methods to complex geometries. Its stability is complicated by the ill-conditioning of the Fourier extension frame. This paper provides a rigorous analysis of the method's pre-asymptotic behavior. We confirm that the spectral collocation system is asymptotically ill-conditioned for both the Poisson and convection-diffusion operators, driven by the redundancy of the underlying frame. However, we prove a fundamental structural dichotomy in their discrete Green's functions. We show that the inverse of the convection-diffusion operator is numerically quasi-sparse, exhibiting exponential off-diagonal decay, in stark contrast to the numerically dense inverse of the Poisson operator. This intrinsic sparsity explains why the convection-diffusion operator is significantly more robust to the underlying frame instability in practical computations.

Figures (5)

• Figure 1: Schematic of the Extended-Domain Method in 1D. The complex physical problem on $I=[0,L]$ is embedded into a larger, regular computational domain $\tilde{I}=[-\delta, L+\delta]$. The solution is approximated by Fourier-like basis functions (e.g., $w_j(x)$) that are defined on the full domain $\tilde{I}$ and vanish at the extended boundaries $-\delta$ and $L+\delta$, but generally do not satisfy boundary conditions at the physical endpoints $0$ and $L$.
• Figure 2: Direct visualization of the structural dichotomy in the physical-space operators and their inverses for $N=64$ and $\delta=2.0$. The colormaps show $\log_{10}$ of the absolute value of the matrix entries. (a), (c): Both forward operators are localized. (b): The inverse of the Poisson operator is dense. (d): The inverse of the convection-diffusion operator is numerically quasi-sparse, visually confirming the exponential decay predicted in Theorem \ref{['thm:exp_decay_cd']}.
• Figure 3: Baseline instability. (a) The Lebesgue constant $\Lambda_N$ grows exponentially for both operators, confirming the underlying frame instability. (b) However, the Convection-Diffusion system maintains a Lebesgue constant orders of magnitude lower than Poisson in the pre-asymptotic regime.
• Figure 4: Visual verification of the Structural Dichotomy (Theorem 4.6). (a) Linear Scale: The Poisson response (Blue) is a global symmetric arch, indicating that the inverse matrix is dense; information at the center affects the entire domain. The Convection-Diffusion response (Orange) is a localized, asymmetric pulse. (b) Log Scale: The upstream side of the Convection-Diffusion response exhibits a straight linear slope, confirming the exponential decay predicted by our theory. This sparsity isolates numerical errors, explaining the method's robustness in convection-dominated regimes.
• Figure 5: 2D Convergence. (a) Poisson. (b) Constant Coeff CD. (c) Variable Coeff CD. The method converges spectrally, demonstrating that the structural robustness extends to tensor-product domains.

Theorems & Definitions (37)

• Definition 2.1: Laplacian Eigenbasis on the Extended Domain
• Definition 2.2: Spectral Collocation Method
• Definition 2.3: Lebesgue Constant
• Remark 2.4: Lebesgue Constant as an Instability Detector
• Theorem 3.1: Exponential Ill-Conditioning of the Poisson System
• proof
• Remark 3.2: The Implication of Square Systems
• Lemma 4.1: Decay of Inverse Entries jaffard1990proprietes
• proof
• Lemma 4.2: Localization of Cardinal Functions