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arXiv · 2508.16060

Using the Immersed Penalized Boundary Method with Splines to Solve PDE's on Curved Domains in 3D

Aussie Greene·Larry L. Schumaker
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Abstract

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.

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Aussie Greene, Larry L. Schumaker. 2025-08-22. Using the Immersed Penalized Boundary Method with Splines to Solve PDE's on Curved Domains in 3D. https://arxiv.org/abs/2508.16060

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