A Singularity Guided Nyström Method for Elastostatics on Two Dimensional Domains with Corners
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914212173840384 |
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| author | Xie, Baoling Lai, Jun |
| author_facet | Xie, Baoling Lai, Jun |
| contents | We develop a comprehensive analytical and numerical framework for boundary integral equations (BIEs) of the 2D Lamé system on cornered domains. By applying local Mellin analysis on a wedge, we obtain a factorizable characteristic equation for the singular exponents of the boundary densities, and clarify their dependence on boundary conditions. The Fredholm well-posedness of the BIEs on cornered domains is proved in weighted Sobolev spaces. We further construct an explicit density-to-Taylor mapping for the BIE and show its invertibility for all but a countable set of angles. Based on these analytical results, we propose a singularity guided Nyström (SGN) scheme for the numerical solution of BIEs on cornered domains. The SGN uses the computed corner exponents and a Legendre-tail indicator to drive panel refinement. An error analysis that combines this refinement strategy with an exponentially accurate far-field quadrature rule is provided. Numerical experiments across various cornered geometries demonstrate that SGN obtains higher order accuracy than uniform Nyström method and reveal a crowding-limited regime for domains with re-entrant angles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18208 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Singularity Guided Nyström Method for Elastostatics on Two Dimensional Domains with Corners Xie, Baoling Lai, Jun Numerical Analysis We develop a comprehensive analytical and numerical framework for boundary integral equations (BIEs) of the 2D Lamé system on cornered domains. By applying local Mellin analysis on a wedge, we obtain a factorizable characteristic equation for the singular exponents of the boundary densities, and clarify their dependence on boundary conditions. The Fredholm well-posedness of the BIEs on cornered domains is proved in weighted Sobolev spaces. We further construct an explicit density-to-Taylor mapping for the BIE and show its invertibility for all but a countable set of angles. Based on these analytical results, we propose a singularity guided Nyström (SGN) scheme for the numerical solution of BIEs on cornered domains. The SGN uses the computed corner exponents and a Legendre-tail indicator to drive panel refinement. An error analysis that combines this refinement strategy with an exponentially accurate far-field quadrature rule is provided. Numerical experiments across various cornered geometries demonstrate that SGN obtains higher order accuracy than uniform Nyström method and reveal a crowding-limited regime for domains with re-entrant angles. |
| title | A Singularity Guided Nyström Method for Elastostatics on Two Dimensional Domains with Corners |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2512.18208 |