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Volume 6, Issue 4
$N$-Simplex Crouzeix-Raviart Element for the Second Order Elliptic/Eigenvalue Problems

Y. Yang, F. Lin & Z. Zhang

Int. J. Numer. Anal. Mod., 6 (2009), pp. 615-626.

Published online: 2009-06

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  • Abstract

We study the $n$-simplex nonconforming Crouzeix-Raviart element in approximating the $n$-dimensional second-order elliptic boundary value problems and the associated eigenvalue problems. By using the second Strang Lemma, optimal rate of convergence is established under the discrete energy norm. The error bound is also valid for the eigenfunction approximations. In addition, when eigenfunctions are singular, we prove that the Crouzeix-Raviart element approximates exact eigenvalues from below. Moreover, our numerical experiments demonstrate that the lower bound property is also valid for smooth eigenfunctions, although a theoretical justification is lacking.

  • Keywords

$n$-simplex, nonconforming Crouzeix-Raviart element, second order elliptic equation, error estimates, eigenvalues, lower bound.

  • AMS Subject Headings

65N25, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{IJNAM-6-615, author = {Y. Yang, F. Lin and Z. Zhang}, title = {$N$-Simplex Crouzeix-Raviart Element for the Second Order Elliptic/Eigenvalue Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2009}, volume = {6}, number = {4}, pages = {615--626}, abstract = {

We study the $n$-simplex nonconforming Crouzeix-Raviart element in approximating the $n$-dimensional second-order elliptic boundary value problems and the associated eigenvalue problems. By using the second Strang Lemma, optimal rate of convergence is established under the discrete energy norm. The error bound is also valid for the eigenfunction approximations. In addition, when eigenfunctions are singular, we prove that the Crouzeix-Raviart element approximates exact eigenvalues from below. Moreover, our numerical experiments demonstrate that the lower bound property is also valid for smooth eigenfunctions, although a theoretical justification is lacking.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/787.html} }
TY - JOUR T1 - $N$-Simplex Crouzeix-Raviart Element for the Second Order Elliptic/Eigenvalue Problems AU - Y. Yang, F. Lin & Z. Zhang JO - International Journal of Numerical Analysis and Modeling VL - 4 SP - 615 EP - 626 PY - 2009 DA - 2009/06 SN - 6 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/787.html KW - $n$-simplex, nonconforming Crouzeix-Raviart element, second order elliptic equation, error estimates, eigenvalues, lower bound. AB -

We study the $n$-simplex nonconforming Crouzeix-Raviart element in approximating the $n$-dimensional second-order elliptic boundary value problems and the associated eigenvalue problems. By using the second Strang Lemma, optimal rate of convergence is established under the discrete energy norm. The error bound is also valid for the eigenfunction approximations. In addition, when eigenfunctions are singular, we prove that the Crouzeix-Raviart element approximates exact eigenvalues from below. Moreover, our numerical experiments demonstrate that the lower bound property is also valid for smooth eigenfunctions, although a theoretical justification is lacking.

Y. Yang, F. Lin and Z. Zhang. (2009). $N$-Simplex Crouzeix-Raviart Element for the Second Order Elliptic/Eigenvalue Problems. International Journal of Numerical Analysis and Modeling. 6 (4). 615-626. doi:
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