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

Po-Yi Wu

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arXiv · 2509.08745

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

Po-Yi Wu

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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 an 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 highly asymmetric, exhibiting exponential upstream decay, in contrast to the numerically dense inverse of the Poisson operator. This intrinsic asymmetry explains why the convection-diffusion operator is significantly more robust to the underlying frame instability in practical computations.

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Po-Yi Wu. 2025-09-10. On the Pre-Asymptotic Stability and Inverse Structure of Extended-Domain Spectral Methods. https://arxiv.org/abs/2509.08745

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