Calculus on Surfaces with General Closest Point Functions

TL;DR

This paper explores the theoretical basis of the Closest Point Method for surface PDEs, introducing a general class of closest point functions, characterizing their properties, and demonstrating their practical effectiveness.

Contribution

It provides a theoretical foundation for the Closest Point Method by defining and characterizing a broad class of closest point functions and validating their use numerically.

Findings

Introduces a general class of closest point functions.

Provides a geometric characterization of these functions.

Numerical experiments demonstrate their effectiveness on surface PDEs.

Abstract

The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study the theoretical foundations of this method. The main idea is that surface differentials of a surface function can be replaced with Cartesian differentials of its closest point extension, i.e., its composition with a closest point function. We introduce a general class of these closest point functions (a subset of differentiable retractions), show that these are exactly the functions necessary to satisfy the above idea, and give a geometric characterization of this class. Finally, we construct some closest point functions and demonstrate their effectiveness numerically on surface PDEs.