Optimal Spectral Approximation in the Overlaps for Generalized Finite Element Methods
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915393405190144 |
|---|---|
| author | Alber, Christian Bastian, Peter Hauck, Moritz Scheichl, Robert |
| author_facet | Alber, Christian Bastian, Peter Hauck, Moritz Scheichl, Robert |
| contents | In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on rings around the boundary of local subdomains. Compared to the corresponding method that solves eigenvalue problems on the whole subdomains, the problem size and the bandwidth of the resulting system matrices are substantially reduced, resulting in faster spectral computations. We prove a nearly exponential a priori decay result for the local approximation errors of the proposed method, which implies the nearly exponential decay of the overall approximation error of the method. The proposed method can also be used as a preconditioner, and only a slight adaptation of our theory is necessary to prove the optimal convergence of the preconditioned iteration. Numerical experiments are presented to support the effectiveness of the proposed method and to investigate its coefficient robustness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Spectral Approximation in the Overlaps for Generalized Finite Element Methods Alber, Christian Bastian, Peter Hauck, Moritz Scheichl, Robert Numerical Analysis 65F10, 65N15, 65N30, 65N55 In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on rings around the boundary of local subdomains. Compared to the corresponding method that solves eigenvalue problems on the whole subdomains, the problem size and the bandwidth of the resulting system matrices are substantially reduced, resulting in faster spectral computations. We prove a nearly exponential a priori decay result for the local approximation errors of the proposed method, which implies the nearly exponential decay of the overall approximation error of the method. The proposed method can also be used as a preconditioner, and only a slight adaptation of our theory is necessary to prove the optimal convergence of the preconditioned iteration. Numerical experiments are presented to support the effectiveness of the proposed method and to investigate its coefficient robustness. |
| title | Optimal Spectral Approximation in the Overlaps for Generalized Finite Element Methods |
| topic | Numerical Analysis 65F10, 65N15, 65N30, 65N55 |
| url | https://arxiv.org/abs/2507.12226 |