A parametrix method for elliptic surface PDEs
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915203194552320 |
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| author | Goodwill, Tristan O'Neil, Michael |
| author_facet | Goodwill, Tristan O'Neil, Michael |
| contents | Elliptic problems along smooth surfaces embedded in three dimensions occur in thin-membrane mechanics, electromagnetics (harmonic vector fields), and computational geometry. In this work, we present a parametrix-based integral equation method applicable to several forms of variable coefficient surface elliptic problems. Via the use of an approximate Green's function, the surface PDEs are transformed into well-conditioned integral equations. We demonstrate high-order numerical examples of this method applied to problems on general surfaces using a variant of the fast multipole method based on smooth interpolation properties of the kernel. Lastly, we discuss extensions of the method to surfaces with boundaries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12501 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A parametrix method for elliptic surface PDEs Goodwill, Tristan O'Neil, Michael Numerical Analysis 35R01, 35C15, 65R20, 35J47, 45B05 Elliptic problems along smooth surfaces embedded in three dimensions occur in thin-membrane mechanics, electromagnetics (harmonic vector fields), and computational geometry. In this work, we present a parametrix-based integral equation method applicable to several forms of variable coefficient surface elliptic problems. Via the use of an approximate Green's function, the surface PDEs are transformed into well-conditioned integral equations. We demonstrate high-order numerical examples of this method applied to problems on general surfaces using a variant of the fast multipole method based on smooth interpolation properties of the kernel. Lastly, we discuss extensions of the method to surfaces with boundaries. |
| title | A parametrix method for elliptic surface PDEs |
| topic | Numerical Analysis 35R01, 35C15, 65R20, 35J47, 45B05 |
| url | https://arxiv.org/abs/2401.12501 |