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Comparison Results of Three First-order Finite Element Methods for the Poisson Model Problem

Please always quote using this URN:urn:nbn:de:0296-matheon-9407
  • This paper establishes the equivalence of conforming Courant finite element method and nonconforming Crouzeix-Raviart finite element method in the sense that the respective energy error norms are equivalent up to generic constants and higher-order data oscillations in a Poisson model problem. The Raviart-Thomas mixed finite element method is better than the previous two whereas the conjecture of the converse relation is proved to be false. This paper completes the analysis of comparison initiated by Braess in Calcolo (2010). Two numerical benchmarks illustrate the comparison theorems and the possible strict superiority of the Raviart-Thomas mixed finite element method. Applications include least-squares finite element methods and equality of approximation classes for concepts of optimality for adaptive finite element methods.

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Metadaten
Author:Carsten Carstensen, Daniel Peterseim, Mira Schedensack
URN:urn:nbn:de:0296-matheon-9407
Referee:Michael Hintermüller
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2011/12/11
Release Date:2011/12/11
Tag:Crouzeix-Raviart; Raviart-Thomas; approximation class; finite element method; least-squares method; nonconforming FEM
Institute:Humboldt-Universität zu Berlin
MSC-Classification:65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N12 Stability and convergence of numerical methods
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
65-XX NUMERICAL ANALYSIS / 65Yxx Computer aspects of numerical algorithms / 65Y20 Complexity and performance of numerical algorithms [See also 68Q25]
Preprint Number:831
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