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arXiv · 1202.3001

Calculus on Surfaces with General Closest Point Functions

Thomas März·Colin B. Macdonald
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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.

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Thomas März, Colin B. Macdonald. 2012-02-14. Calculus on Surfaces with General Closest Point Functions. https://doi.org/10.1137/120865537

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