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The scaling and skewness of optimally transported meshes on the sphere

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Title: The scaling and skewness of optimally transported meshes on the sphere
Authors: Budd, CJ
McRae, ATT
Cotter, CJ
Item Type: Journal Article
Abstract: In the context of numerical solution of PDEs, dynamic mesh redistribution methods (r-adaptive methods) are an important procedure for increasing the resolution in regions of interest, without modifying the connectivity of the mesh. Key to the success of these methods is that the mesh should be sufficiently refined (locally) and flexible in order to resolve evolving solution features, but at the same time not introduce errors through skewness and lack of regularity. Some state-of-the-art methods are bottom-up in that they attempt to prescribe both the local cell size and the alignment to features of the solution. However, the resulting problem is overdetermined, necessitating a compromise between these conflicting requirements. An alternative approach, described in this paper, is to prescribe only the local cell size and augment this an optimal transport condition to provide global regularity. This leads to a robust and flexible algorithm for generating meshes fitted to an evolving solution, with minimal need for tuning parameters. Of particular interest for geophysical modelling are meshes constructed on the surface of the sphere. The purpose of this paper is to demonstrate that meshes generated on the sphere using this optimal transport approach have good a-priori regularity and that the meshes produced are naturally aligned to various simple features. It is further shown that the sphere's intrinsic curvature leads to more regular meshes than the plane. In addition to these general results, we provide a wide range of examples relevant to practical applications, to showcase the behaviour of optimally transported meshes on the sphere. These range from axisymmetric cases that can be solved analytically to more general examples that are tackled numerically. Evaluation of the singular values and singular vectors of the mesh transformation provides a quantitative measure of the mesh anisotropy, and this is shown to match analytic predictions.
Issue Date: 15-Dec-2018
Date of Acceptance: 18-Aug-2018
URI: http://hdl.handle.net/10044/1/63572
DOI: https://dx.doi.org/10.1016/j.jcp.2018.08.028
ISSN: 0021-9991
Start Page: 540
End Page: 564
Journal / Book Title: Journal of Computational Physics
Volume: 375
Copyright Statement: © 2018 Published by Elsevier Inc. All rights reserved. This manuscript is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International Licence http://creativecommons.org/licenses/by-nc-nd/4.0/
Sponsor/Funder: Natural Environment Research Council (NERC)
Funder's Grant Number: NE/M013634/1
Keywords: 01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Applied Mathematics
Publication Status: Published
Online Publication Date: 2018-08-31
Appears in Collections:Mathematics
Applied Mathematics and Mathematical Physics
Faculty of Natural Sciences



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