CONTINUOUS MESH FRAMEWORK PART I: WELL-POSED CONTINUOUS INTERPOLATION ERROR

In the context of mesh adaptation, Riemannian metric spaces have been used to prescribe orientation, density, and stretching of anisotropic meshes. But, such structures are only considered to compute lengths in adaptive mesh generators. In this article, a Riemannian metric space is shown to be more...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2011-01, Vol.49 (1/2), p.38-60
Main Authors: LOSEILLE, ADRIEN, ALAUZET, FRÉDÉRIC
Format: Article
Language:English
Subjects:
Adaptation
Analysis of PDEs
Approximations and expansions
Calculus of variations and optimal control
Density
Error analysis
Errors
Euclidean space
Exact sciences and technology
Finite element method
General topology
Interpolation
Linear interpolation
Mathematical analysis
Mathematical duality
Mathematical functions
Mathematical models
Mathematics
Metric space
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Sciences and techniques of general use
Studies
Symmetry
Tensors
Tetrahedrons
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Unit ball
Vertices
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Summary:In the context of mesh adaptation, Riemannian metric spaces have been used to prescribe orientation, density, and stretching of anisotropic meshes. But, such structures are only considered to compute lengths in adaptive mesh generators. In this article, a Riemannian metric space is shown to be more than a way to compute a length. It is proven to be a reliable continuous mesh model. In particular, we demonstrate that the linear interpolation error can be evaluated continuously on a Riemannian metric space. From one hand, this new continuous framework proves that prescribing a Riemannian metric field is equivalent to the local control in the L 1 norm of the interpolation error. This proves the consistency of classical metric-based mesh adaptation procedures. On the other hand, powerful mathematical tools are available and are well defined on Riemannian metric spaces: calculus of variations, differentiation, optimization, ..., whereas these tools are not defined on discrete meshes.
ISSN:0036-1429
1095-7170
DOI:10.1137/090754078