Using the Immersed Penalized Boundary Method with Splines to Solve PDE's on Curved Domains in 3D
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915456739180544 |
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| author | Greene, Aussie Schumaker, Larry L. |
| author_facet | Greene, Aussie Schumaker, Larry L. |
| contents | Second-order elliptic boundary-value problems defined on curved domains in 2D and 3D arise frequently in practice. A lot of work has gone into developing numerical methods for solving such problems. One of the newest and most promising methods is the $\textit{immersed penalized boundary method}$ (IPBM) introduced in [Schumaker, L. L., Solving elliptic PDE's on domains with curved boundaries with an immersed penalized boundary method, J. Sci. Comp. ${\bf 80(3)}$ (2019), 1369--1394]. For a comprehensive discussion of the use of these methods with various bivariate spline spaces, see the recent book [Schumaker, L. L.: $\textit{Spline Functions: More Computational Methods}$, SIAM (Philadelphia), 2024]. The purpose of this paper is to show how to use IPBM methods with trivariate spline spaces to solve boundary-value problems on curved domains in 3D. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16060 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Using the Immersed Penalized Boundary Method with Splines to Solve PDE's on Curved Domains in 3D Greene, Aussie Schumaker, Larry L. Numerical Analysis 65D07 (Primary), 65N99 (Secondary) Second-order elliptic boundary-value problems defined on curved domains in 2D and 3D arise frequently in practice. A lot of work has gone into developing numerical methods for solving such problems. One of the newest and most promising methods is the $\textit{immersed penalized boundary method}$ (IPBM) introduced in [Schumaker, L. L., Solving elliptic PDE's on domains with curved boundaries with an immersed penalized boundary method, J. Sci. Comp. ${\bf 80(3)}$ (2019), 1369--1394]. For a comprehensive discussion of the use of these methods with various bivariate spline spaces, see the recent book [Schumaker, L. L.: $\textit{Spline Functions: More Computational Methods}$, SIAM (Philadelphia), 2024]. The purpose of this paper is to show how to use IPBM methods with trivariate spline spaces to solve boundary-value problems on curved domains in 3D. |
| title | Using the Immersed Penalized Boundary Method with Splines to Solve PDE's on Curved Domains in 3D |
| topic | Numerical Analysis 65D07 (Primary), 65N99 (Secondary) |
| url | https://arxiv.org/abs/2508.16060 |