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

Le résumé fourni par la source

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.

Ce résumé expose les affirmations des auteurs. BNTIC ne l’interprète pas comme une validation indépendante des résultats.