28 Citations (SciVal)
205 Downloads (Pure)

Abstract

In moving mesh methods, the underlying mesh is dynamically adapted without changing the connectivity of the mesh. We specifically consider the generation of meshes which are adapted to a scalar monitor function through equidistribution. Together with an optimal-transport condition, this leads to a Monge-Amp ere equation for a scalar mesh potential. We adapt an existing finite element scheme for the standard Monge-Amp ere equation to this mesh generation problem; this is a mixed finite element scheme, in which an extra discrete variable is introduced to represent the Hessian matrix of second derivatives. The problem we consider has additional nonlinearities over the basic Monge-Amp ere equation due to the implicit dependence of the monitor function on the resulting mesh. We also derive an equivalent Monge-Amp ere-like equation for generating meshes on the sphere. The finite element scheme is extended to the sphere, and we provide numerical examples. All numerical experiments are performed using the open-source finite element framework Firedrake.

Original languageEnglish
Pages (from-to)A1121-A1148
JournalSIAM Journal on Scientific Computing
Volume40
Issue number2
Early online date24 Apr 2018
DOIs
Publication statusPublished - 24 Apr 2018

Funding

\ast Submitted to the journal's Methods and Algorithms for Scientific Computing section December 27, 2016; accepted for publication (in revised form) January 16, 2018; published electronically April 24, 2018. http://www.siam.org/journals/sisc/40-2/M110951.html Funding: This work was supported by the Natural Environment Research Council under grants NE/M013480/1 and NE/M013634/1. \dagger Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK (andrew.mcrae@ physics.ox.ac.uk, [email protected]). \ddagger Department of Mathematics, Imperial College London, London, SW7 2AZ, UK (colin.cotter@ imperial.ac.uk).

Keywords

  • Finite element
  • Mesh adaptivity
  • Monge-Amp ere equation
  • Optimal transport

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Fingerprint

Dive into the research topics of 'Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements'. Together they form a unique fingerprint.

Cite this