Sharp Ascent--Descent Spectral Stability under Strong Resolvent Convergence
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2025
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| author | Ennaceur, Marwa |
| author_facet | Ennaceur, Marwa |
| contents | We establish sharp stability results for of non--selfadjoint the ascent and descent spectra under strong resolvent convergence (SRS), a natural framework for finite element approximations of non-selfadjoint and singularly perturbed operators. The key quantitative hypothesis is the reduced minimum modulus $γ(T-λ)>0$, which guarantees closed range and enables the transfer of the Kaashoek -- Taylor criteria via gap convergence of operator graphs. At the essential level, B--Fredholm theory extends stability to powers $(T-λ)^m$ provided $γ((T-λ)^j)>0$ for all $1\le j\le m$. We introduce a computable finite-element diagnostic $γ_h = σ_{\min}(M^{-1/2}(A_h-λM)M^{-1/2})$, which serves as a practical surrogate for $γ(T-λ)$ and remains uniformly positive even in convection-dominated regimes when stabilized schemes (e.g., SUPG) are employed. Numerical experiments confirm that $\liminf_{h\to0}γ_h>0$ is both necessary and sufficient for spectral stability, while a Volterra-type counterexample demonstrates the indispensability of the closed-range condition for powers. The analysis clarifies why norm resolvent convergence fails for rough or singular limits, and how SRS-combined with quantitative control of $γ_h$--rescues ascent--descent stability in realistic computational settings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_20971 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp Ascent--Descent Spectral Stability under Strong Resolvent Convergence Ennaceur, Marwa Numerical Analysis Functional Analysis Spectral Theory 47A10, 47A53, 47A55, 47A58, 47N40, 65N30, 35P05 G.1.8; G.1.3; F.2.1 We establish sharp stability results for of non--selfadjoint the ascent and descent spectra under strong resolvent convergence (SRS), a natural framework for finite element approximations of non-selfadjoint and singularly perturbed operators. The key quantitative hypothesis is the reduced minimum modulus $γ(T-λ)>0$, which guarantees closed range and enables the transfer of the Kaashoek -- Taylor criteria via gap convergence of operator graphs. At the essential level, B--Fredholm theory extends stability to powers $(T-λ)^m$ provided $γ((T-λ)^j)>0$ for all $1\le j\le m$. We introduce a computable finite-element diagnostic $γ_h = σ_{\min}(M^{-1/2}(A_h-λM)M^{-1/2})$, which serves as a practical surrogate for $γ(T-λ)$ and remains uniformly positive even in convection-dominated regimes when stabilized schemes (e.g., SUPG) are employed. Numerical experiments confirm that $\liminf_{h\to0}γ_h>0$ is both necessary and sufficient for spectral stability, while a Volterra-type counterexample demonstrates the indispensability of the closed-range condition for powers. The analysis clarifies why norm resolvent convergence fails for rough or singular limits, and how SRS-combined with quantitative control of $γ_h$--rescues ascent--descent stability in realistic computational settings. |
| title | Sharp Ascent--Descent Spectral Stability under Strong Resolvent Convergence |
| topic | Numerical Analysis Functional Analysis Spectral Theory 47A10, 47A53, 47A55, 47A58, 47N40, 65N30, 35P05 G.1.8; G.1.3; F.2.1 |
| url | https://arxiv.org/abs/2511.20971 |