Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.08745 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914202325614592 |
|---|---|
| author | Wu, Po-Yi |
| author_facet | Wu, Po-Yi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08745 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Pre-Asymptotic Stability and Inverse Structure of Extended-Domain Spectral Methods Wu, Po-Yi Numerical Analysis 65N12, 65N35, 65T40 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. |
| title | On the Pre-Asymptotic Stability and Inverse Structure of Extended-Domain Spectral Methods |
| topic | Numerical Analysis 65N12, 65N35, 65T40 |
| url | https://arxiv.org/abs/2509.08745 |