1. A family of Crouzeix–Raviart finite elements in 3D
- Author
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Charles F. Dunkl, Stefan A. Sauter, Patrick Ciarlet, University of Zurich, Sauter, Stefan A, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Departement of Mathematics [Charlottesville], University of Virginia [Charlottesville], Institut für Mathematik [Zürich], Universität Zürich [Zürich] = University of Zurich (UZH), and University of Virginia more...
- Subjects
Pure mathematics ,Crouzeix-Raviart ,Basis function ,010103 numerical & computational mathematics ,Type (model theory) ,Space (mathematics) ,01 natural sciences ,510 Mathematics ,Finite element ,2604 Applied Mathematics ,FOS: Mathematics ,Orthogonal polynomials on triangles ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics ,Basis (linear algebra) ,Symmetric orthogonal polynomials ,Applied Mathematics ,2603 Analysis ,010102 general mathematics ,Numerical Analysis (math.NA) ,Primary 33C45, 33C50, 65N12, 65N30, Secondary 33C80 ,Finite element method ,10123 Institute of Mathematics ,Non-conforming ,Orthogonal polynomials ,Jump ,Linear independence ,AMS-Classification : 33C45, 33C50, 65N12, 65N30 ,secondary 33C80 ,[MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] ,Analysis - Abstract
In this paper we will develop a family of non-conforming "Crouzeix-Raviart" type finite elements in three dimensions. They consist of local polynomials of maximal degree $p\in\mathbb{N}$ on simplicial finite element meshes while certain jump conditions are imposed across adjacent simplices. We will prove optimal a priori estimates for these finite elements. The characterization of this space via jump conditions is implicit and the derivation of a local basis requires some deeper theoretical tools from orthogonal polynomials on triangles and their representation. We will derive these tools for this purpose. These results allow us to give explicit representations of the local basis functions. Finally we will analyze the linear independence of these sets of functions and discuss the question whether they span the whole non-conforming space., Comment: 29 figures more...
- Published
- 2018
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