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 $\mathit{\text{immersed}}\mathit{\text{penalized}}\mathit{\text{boundary}}\mathit{\text{method}}$ (IPBM) introduced in [Schumaker, L. L., Solving elliptic PDE’s on domains with curved boundaries with an immersed penalized boundary method, J. Sci. Comp. $\mathbf{80}(\mathbf{3})$ (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.: $\mathit{\text{Spline}}\mathit{\text{Functions:}}\mathit{\text{More}}\mathit{\text{Computational}}\mathit{\text{Methods}}$, 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.