A Second-order method on graded meshes for fractional Laplacian via Riesz fractional derivative with a singular source term
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915154230247424 |
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| author | Chen, Minghua Han, Jianxing Shi, Jiankang Yu, Fan |
| author_facet | Chen, Minghua Han, Jianxing Shi, Jiankang Yu, Fan |
| contents | The high-order numerical analysis for fractional Laplacian via the Riesz fractional derivative, under the low regularity solution, has presented significant challenges in the past decades. To fill in this gap, we design a grid mapping function on graded meshes to analyse the local truncation errors, which are far less than second-order convergence at the boundary layer. To restore the second-order global errors, we construct an appropriate right-preconditioner for the resulting matrix algebraic equation. We prove that the proposed scheme achieves second-order convergence on graded meshes even if the source term is singular or hypersingular. Numerical experiments illustrate the theoretical results. The proposed approach is applicable for multidimensional fractional diffusion equations, gradient flows and nonlinear equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_11117 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Second-order method on graded meshes for fractional Laplacian via Riesz fractional derivative with a singular source term Chen, Minghua Han, Jianxing Shi, Jiankang Yu, Fan Numerical Analysis 26A33, 26A30, 65L20 The high-order numerical analysis for fractional Laplacian via the Riesz fractional derivative, under the low regularity solution, has presented significant challenges in the past decades. To fill in this gap, we design a grid mapping function on graded meshes to analyse the local truncation errors, which are far less than second-order convergence at the boundary layer. To restore the second-order global errors, we construct an appropriate right-preconditioner for the resulting matrix algebraic equation. We prove that the proposed scheme achieves second-order convergence on graded meshes even if the source term is singular or hypersingular. Numerical experiments illustrate the theoretical results. The proposed approach is applicable for multidimensional fractional diffusion equations, gradient flows and nonlinear equations. |
| title | A Second-order method on graded meshes for fractional Laplacian via Riesz fractional derivative with a singular source term |
| topic | Numerical Analysis 26A33, 26A30, 65L20 |
| url | https://arxiv.org/abs/2502.11117 |