Implementation of Worsey-Farin splines with applications to solution transfer

TL;DR

This paper provides comprehensive implementation details for Worsey-Farin spline interpolation on tetrahedral meshes, demonstrating its effectiveness in solution transfer applications and analyzing accuracy, conservation, and visualization properties.

Contribution

It offers the first detailed implementation guide for WF-splines and evaluates their performance in solution transfer, including error estimates and conservation improvements.

Findings

WF-spline implementation details are now available for practical use.

Solution transfer using WF-splines is accurate and conservative with adaptive quadrature.

Numerical experiments confirm the effectiveness of the proposed methods.

Abstract

This work primarily focuses on providing full implementation details for Worsey-Farin (WF) spline interpolation over tetrahedral elements. While this spline space is not new and the theory has been covered in other works, there is a lack of explicit and comprehensive implementation details, which we hope to provide. In this paper, we also demonstrate the effectiveness of the WF-spline space through a simple target application: solution transfer. Moreover, we derive an error estimate for the WF spline-based, solution transfer process. We conduct numerical experiments quantifying the conservative nature and order of accuracy of the transfer process, and we present a qualitative evaluation of the visualization properties of the smoothed solution. Additionally, in our study of conservation, we demonstrate how adaptive numerical quadrature rules on the tetrahedron used in conjunction with global L2-projection can improve the conservation of the solution transfer process.