Projection method for quasiperiodic elliptic equations and application to quasiperiodic homogenization
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915238767493120 |
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| author | Jiang, Kai Li, Meng Zhang, Juan Zhang, Lei |
| author_facet | Jiang, Kai Li, Meng Zhang, Juan Zhang, Lei |
| contents | In this study, we address the challenge of solving elliptic equations with quasiperiodic coefficients. To achieve accurate and efficient computation, we introduce the projection method, which enables the embedding of quasiperiodic systems into higher-dimensional periodic systems. To enhance the computational efficiency, we propose a compressed storage strategy for the stiffness matrix by its multi-level block circulant structure, significantly reducing memory requirements. Furthermore, we design a diagonal preconditioner to efficiently solve the resulting high-dimensional linear system by reducing the condition number of the stiffness matrix. These techniques collectively contribute to the computational effectiveness of our proposed approach. Convergence analysis shows the polynomial accuracy of the proposed method. We demonstrate the effectiveness and accuracy of our approach through a series of numerical examples. Moreover, we apply our method to achieve a highly accurate computation of the homogenized coefficients for a quasiperiodic multiscale elliptic equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06841 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projection method for quasiperiodic elliptic equations and application to quasiperiodic homogenization Jiang, Kai Li, Meng Zhang, Juan Zhang, Lei Numerical Analysis 65D05, 65D15, 65F08, 35B15, 35B27 In this study, we address the challenge of solving elliptic equations with quasiperiodic coefficients. To achieve accurate and efficient computation, we introduce the projection method, which enables the embedding of quasiperiodic systems into higher-dimensional periodic systems. To enhance the computational efficiency, we propose a compressed storage strategy for the stiffness matrix by its multi-level block circulant structure, significantly reducing memory requirements. Furthermore, we design a diagonal preconditioner to efficiently solve the resulting high-dimensional linear system by reducing the condition number of the stiffness matrix. These techniques collectively contribute to the computational effectiveness of our proposed approach. Convergence analysis shows the polynomial accuracy of the proposed method. We demonstrate the effectiveness and accuracy of our approach through a series of numerical examples. Moreover, we apply our method to achieve a highly accurate computation of the homogenized coefficients for a quasiperiodic multiscale elliptic equation. |
| title | Projection method for quasiperiodic elliptic equations and application to quasiperiodic homogenization |
| topic | Numerical Analysis 65D05, 65D15, 65F08, 35B15, 35B27 |
| url | https://arxiv.org/abs/2404.06841 |