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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Veröffentlicht in:SIAM journal on numerical analysis Jg. 49; H. 1/2; S. 38 - 60
Hauptverfasser: LOSEILLE, ADRIEN, ALAUZET, FRÉDÉRIC
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
Schlagworte:
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
65N15
Linear interpolation
Grid pattern
Metric space
Error estimation
Optimization method
Riemannian metric space
anisotropic mesh adaptation
interpolation error
Numerical approximation
optimal interpolation error bound
Numerical analysis
linear interpolate
65D05
65N50
Variational calculus
65L50
unstructured mesh
Mathematical programming
continuous mesh
optimal interpolation error bound AMS subject classifications. 65D05
ISSN:0036-1429, 1095-7170
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Zusammenfassung: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.
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ISSN:0036-1429
1095-7170
DOI:10.1137/090754078