A SIMPLE NECESSARY AND SUFFICIENT CONDITION FOR THE ENRICHMENT OF THE CROUZEIX-RAVIART ELEMENT

We provide a simple condition, which is both necessary and sufficient, that guarantees the existence of an enriched Crouzeix-Raviart element. Our main result shows that the latter can be easily expressed in terms of the approximation error in a multivariate generalized trapezoidal type cubature form...

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Bibliographic Details
Published in:Applicable analysis and discrete mathematics 2016, Vol.10 (2), p.378-393
Main Authors: Bachar, Mostafa, Guessab, Allal
Format: Article
Language:English
Subjects:
Analysis of PDEs
Approximation
Classical Analysis and ODEs
Computer Science
Coordinate systems
Degrees of freedom
Differential Geometry
Discrete Mathematics
Human error
Logical givens
Mathematical functions
Mathematics
Numerical Analysis
Polynomials
Real numbers
Sufficient conditions
Vertices
Citations: Items that cite this one
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Summary:We provide a simple condition, which is both necessary and sufficient, that guarantees the existence of an enriched Crouzeix-Raviart element. Our main result shows that the latter can be easily expressed in terms of the approximation error in a multivariate generalized trapezoidal type cubature formula. Furthermore, we derive simple explicit formulas for its associated basis functions, and then prove how to use them to characterize all admissible added degrees of freedom, that generate well defined enriched Crouzeix-Raviart elements. We also show that the approximation error using the proposed enriched element can be written as the error of the (non-enriched) Crouzeix- Raviart element plus a perturbation that depends on the enrichment function. Finally, we estimate the approximation error in L2 norm, with explicit constants in both two and three dimensions. A complement to this result is also given for any dimension.
ISSN:1452-8630
2406-100X
DOI:10.2298/AADM160610012B