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Volume 14, Issue 3
Nonconforming Finite Volume Methods for Second Order Elliptic Boundary Value Problems

Yuanyuan Zhang & Zhongying Cheng

Int. J. Numer. Anal. Mod., 14 (2017), pp. 381-404.

Published online: 2017-06

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  • Abstract

This paper is devoted to the analysis of nonconforming finite volume methods (FVMs), whose trial spaces are chosen as the nonconforming finite element (FE) spaces, for solving the second order elliptic boundary value problems. We formulate the nonconforming FVMs as special types of Petrov-Galerkin methods and develop a general convergence theorem, which serves as a guide for the analysis of the nonconforming FVMs. As special examples, we shall present the triangulation based Crouzeix-Raviart (C-R) FVM as well as the rectangle mesh based hybrid Wilson FVM. Their optimal error estimates in the mesh dependent $H^1$-norm will be obtained under the condition that the primary mesh is regular. For the hybrid Wilson FVM, we prove that it enjoys the same optimal error order in the $L^2$-norm as that of the Wilson FEM. Numerical experiments are also presented to confirm the theoretical results.

  • Keywords

Nonconforming finite volume method, elliptic boundary value problems.

  • AMS Subject Headings

65N30, 65N12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-14-381, author = {Yuanyuan Zhang and Zhongying Cheng}, title = {Nonconforming Finite Volume Methods for Second Order Elliptic Boundary Value Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2017}, volume = {14}, number = {3}, pages = {381--404}, abstract = {

This paper is devoted to the analysis of nonconforming finite volume methods (FVMs), whose trial spaces are chosen as the nonconforming finite element (FE) spaces, for solving the second order elliptic boundary value problems. We formulate the nonconforming FVMs as special types of Petrov-Galerkin methods and develop a general convergence theorem, which serves as a guide for the analysis of the nonconforming FVMs. As special examples, we shall present the triangulation based Crouzeix-Raviart (C-R) FVM as well as the rectangle mesh based hybrid Wilson FVM. Their optimal error estimates in the mesh dependent $H^1$-norm will be obtained under the condition that the primary mesh is regular. For the hybrid Wilson FVM, we prove that it enjoys the same optimal error order in the $L^2$-norm as that of the Wilson FEM. Numerical experiments are also presented to confirm the theoretical results.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10013.html} }
TY - JOUR T1 - Nonconforming Finite Volume Methods for Second Order Elliptic Boundary Value Problems AU - Yuanyuan Zhang & Zhongying Cheng JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 381 EP - 404 PY - 2017 DA - 2017/06 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10013.html KW - Nonconforming finite volume method, elliptic boundary value problems. AB -

This paper is devoted to the analysis of nonconforming finite volume methods (FVMs), whose trial spaces are chosen as the nonconforming finite element (FE) spaces, for solving the second order elliptic boundary value problems. We formulate the nonconforming FVMs as special types of Petrov-Galerkin methods and develop a general convergence theorem, which serves as a guide for the analysis of the nonconforming FVMs. As special examples, we shall present the triangulation based Crouzeix-Raviart (C-R) FVM as well as the rectangle mesh based hybrid Wilson FVM. Their optimal error estimates in the mesh dependent $H^1$-norm will be obtained under the condition that the primary mesh is regular. For the hybrid Wilson FVM, we prove that it enjoys the same optimal error order in the $L^2$-norm as that of the Wilson FEM. Numerical experiments are also presented to confirm the theoretical results.

Yuanyuan Zhang and Zhongying Cheng. (2017). Nonconforming Finite Volume Methods for Second Order Elliptic Boundary Value Problems. International Journal of Numerical Analysis and Modeling. 14 (3). 381-404. doi:
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