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

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.