Calculus on Surfaces with General Closest Point Functions

Thomas März, Colin B. Macdonald

Research output: Contribution to journalArticlepeer-review

37 Scopus citations

Abstract

The closest point method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys., 227 (2008), pp. 1943- 1961] 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. © 2012 Society for Industrial and Applied Mathematics.
Original languageEnglish (US)
Pages (from-to)3303-3328
Number of pages26
JournalSIAM Journal on Numerical Analysis
Volume50
Issue number6
DOIs
StatePublished - Jan 2012
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: Received by the editors February 10, 2012; accepted for publication (in revised form) September 24, 2012; published electronically December 4, 2012. This work was supported by award KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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