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

TL;DR

This paper extends the immersed penalized boundary method (IPBM) using splines to effectively solve second-order elliptic PDEs on curved 3D domains, enhancing numerical solutions for complex geometries.

Contribution

It introduces a novel approach combining IPBM with trivariate spline spaces for 3D curved domain PDEs, advancing computational techniques in this area.

Findings

Demonstrates the effectiveness of IPBM with splines in 3D

Provides numerical results on curved domains

Enhances accuracy of PDE solutions on complex geometries

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.