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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 optimaltransport condition, this leads to a MongeAmp ere equation for a scalar mesh potential. We adapt an existing finite element scheme for the standard MongeAmp 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 MongeAmp ere equation due to the implicit dependence of the monitor function on the resulting mesh. We also derive an equivalent MongeAmp erelike 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 opensource finite element framework Firedrake.
Original language  English 

Pages (fromto)  A1121A1148 
Journal  SIAM Journal on Scientific Computing 
Volume  40 
Issue number  2 
Early online date  24 Apr 2018 
DOIs  
Publication status  Published  24 Apr 2018 
Keywords
 Finite element
 Mesh adaptivity
 MongeAmp ere equation
 Optimal transport
ASJC Scopus subject areas
 Computational Mathematics
 Applied Mathematics
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Dive into the research topics of 'Optimaltransportbased mesh adaptivity on the plane and sphere using finite elements'. Together they form a unique fingerprint.Projects
 1 Finished

Moving Meshes for Global Atmospheric Modelling
Natural Environment Research Council
1/09/15 → 31/08/18
Project: Research council
Profiles

Chris Budd
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Probability Laboratory at Bath
 Centre for Doctoral Training in Decarbonisation of the Built Environment (dCarb)
 Centre for Mathematical Biology
 Institute for Mathematical Innovation (IMI)
 Centre for Nonlinear Mechanics
 EPSRC Centre for Doctoral Training in Advanced Automotive Propulsion Systems (AAPS CDT)
Person: Research & Teaching, Affiliate staff