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On the mesh nonsingularity of the moving mesh PDE method

Authors: Weizhang Huang and Lennard Kamenski
Journal: Math. Comp. 87 (2018), 1887-1911
MSC (2010): Primary 65N50, 65K10
Published electronically: October 2, 2017
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Abstract: The moving mesh PDE (MMPDE) method for variational mesh generation and adaptation is studied theoretically at the discrete level, in particular the nonsingularity of the obtained meshes. Meshing functionals are discretized geometrically and the MMPDE is formulated as a modified gradient system of the corresponding discrete functionals for the location of mesh vertices. It is shown that if the meshing functional satisfies a coercivity condition, then the mesh of the semi-discrete MMPDE is nonsingular for all time if it is nonsingular initially. Moreover, the altitudes and volumes of its elements are bounded below by positive numbers depending only on the number of elements, the metric tensor, and the initial mesh. Furthermore, the value of the discrete meshing functional is convergent as time increases, which can be used as a stopping criterion in computation. Finally, the mesh trajectory has limiting meshes which are critical points of the discrete functional. The convergence of the mesh trajectory can be guaranteed when a stronger condition is placed on the meshing functional. Two meshing functionals based on alignment and equidistribution are known to satisfy the coercivity condition. The results also hold for fully discrete systems of the MMPDE provided that the time step is sufficiently small and a numerical scheme preserving the property of monotonically decreasing energy is used for the temporal discretization of the semi-discrete MMPDE. Numerical examples are presented.

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Additional Information

Weizhang Huang
Affiliation: Department of Mathematics, The University of Kansas, Lawrence, Kansas 66045

Lennard Kamenski
Affiliation: Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany

Keywords: Variational mesh generation, mesh adaptation, moving mesh PDE, mesh nonsingularity, limiting mesh
Received by editor(s): December 15, 2015
Received by editor(s) in revised form: February 3, 2017
Published electronically: October 2, 2017
Additional Notes: The first author was supported by the University of Kansas General Research Fund allocation #2301056.
Article copyright: © Copyright 2017 American Mathematical Society

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