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The geometry of r-adaptive meshes generated using optimal transport methods

Walsh, E.; Budd, C. J.; Russell, R. D.; Walsh, E.J.


Emily Walsh
Associate Director Mathematics & Statistics

C. J. Budd

R. D. Russell

Emily Walsh
Associate Director Mathematics & Statistics


© 2014 Elsevier Inc. The principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods, and a metric tensor and mesh metric are useful theoretical tools for understanding a method's level of mesh alignment, or anisotropy. We consider a mesh redistribution method based on the Monge-Ampère equation which combines equidistribution of a given scalar density function with optimal transport. It does not involve explicit use of a metric tensor, although such a tensor must exist for the method, and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh. For model problems with a linear feature and with a radially symmetric feature, we derive the exact form of the metric, which involves expressions for its eigenvalues and eigenvectors. The eigenvectors are shown to be orthogonal and tangential to the feature, and the ratio of the eigenvalues (corresponding to the level of anisotropy) is shown to depend, both locally and globally, on the value of the density function and the amount of curvature. We thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function. Numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features, including for a non-trivial time dependant nonlinear PDE which evolves narrow and curved reaction fronts.


Walsh, E., Budd, C. J., Russell, R. D., & Walsh, E. (2015). The geometry of r-adaptive meshes generated using optimal transport methods. Journal of Computational Physics, 282, 113-137.

Journal Article Type Article
Acceptance Date Nov 4, 2014
Online Publication Date Nov 12, 2014
Publication Date Feb 1, 2015
Journal Journal of Computational Physics
Print ISSN 0021-9991
Electronic ISSN 1090-2716
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 282
Pages 113-137
Keywords alignment, anisotropy, mesh adaptation, metric tensor, Monge–Ampère
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