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

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.