Volume 6, Issue 4
N-simplex Crouzeix-Riviart Element for the Second Order Elliptic/eigenvalue Problems

Y. Yang, F. Lin & Z. Zhang

DOI:

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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@Article{IJNAM-6-615, author = {Y. Yang, F. Lin and Z. Zhang}, title = {N-simplex Crouzeix-Riviart 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-Riviart 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://dor.org/ UR - https://global-sci.org/intro/ijnam/787.html KW - n-simplex KW - nonconforming Crouzeix-Raviart element KW - second order elliptic equation KW - error estimates KW - eigenvalues KW - 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 & Z. Zhang. (1970). N-simplex Crouzeix-Riviart Element for the Second Order Elliptic/eigenvalue Problems. International Journal of Numerical Analysis and Modeling. 6 (4). 615-626. doi:
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