Volume 27, Issue 6
Two-grid Discretization Schemes of the Nonconforming FEM for Eigenvalue Problems

Yidu Yang

J. Comp. Math., 27 (2009), pp. 748-763.

Published online: 2009-12

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  • Abstract

This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou (Math. Comput., 70 (2001), pp.17-25) to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the fine mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming finite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the first eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the fine mesh directly.  

  • Keywords

Nonconforming finite elements Rayleigh quotient Two-grid schemes The lower bounds of eigenvalue High accuracy

  • AMS Subject Headings

65N25 65N30.

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COPYRIGHT: © Global Science Press

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@Article{JCM-27-748, author = {}, title = {Two-grid Discretization Schemes of the Nonconforming FEM for Eigenvalue Problems}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {6}, pages = {748--763}, abstract = {

This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou (Math. Comput., 70 (2001), pp.17-25) to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the fine mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming finite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the first eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the fine mesh directly.  

}, issn = {1991-7139}, doi = {https://doi.org/10.4208//jcm.2009.09-m2876}, url = {http://global-sci.org/intro/article_detail/jcm/8601.html} }
TY - JOUR T1 - Two-grid Discretization Schemes of the Nonconforming FEM for Eigenvalue Problems JO - Journal of Computational Mathematics VL - 6 SP - 748 EP - 763 PY - 2009 DA - 2009/12 SN - 27 DO - http://doi.org/10.4208//jcm.2009.09-m2876 UR - https://global-sci.org/intro/article_detail/jcm/8601.html KW - Nonconforming finite elements KW - Rayleigh quotient KW - Two-grid schemes KW - The lower bounds of eigenvalue KW - High accuracy AB -

This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou (Math. Comput., 70 (2001), pp.17-25) to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the fine mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming finite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the first eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the fine mesh directly.  

Yidu Yang. (2019). Two-grid Discretization Schemes of the Nonconforming FEM for Eigenvalue Problems. Journal of Computational Mathematics. 27 (6). 748-763. doi:10.4208//jcm.2009.09-m2876
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