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Volume 3, Issue 2
$H^1$-Stability and Convergence of the FE, FV and FD Methods for an Elliptic Equation

Yinnian He & Xinlong Feng

DOI: 10.4208/eajam.030513.200513a

East Asian J. Appl. Math., 3 (2013), pp. 154-170.

Published online: 2018-02

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  • Abstract

We obtain the coefficient matrices of the finite element (FE), finite volume (FV) and finite difference (FD) methods based on $P_1$-conforming elements on a quasi-uniform mesh, in order to approximately solve a boundary value problem involving the elliptic Poisson equation. The three methods are shown to possess the same $H^1$-stability and convergence. Some numerical tests are made, to compare the numerical results from the three methods and to review our theoretical results.

  • Keywords

Finite element method, finite difference method, finite volume method, Poisson equation, stability and convergence.

  • AMS Subject Headings

35Q30, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-3-154, author = {Yinnian He and Xinlong Feng}, title = {$H^1$-Stability and Convergence of the FE, FV and FD Methods for an Elliptic Equation}, journal = {East Asian Journal on Applied Mathematics}, year = {2018}, volume = {3}, number = {2}, pages = {154--170}, abstract = {

We obtain the coefficient matrices of the finite element (FE), finite volume (FV) and finite difference (FD) methods based on $P_1$-conforming elements on a quasi-uniform mesh, in order to approximately solve a boundary value problem involving the elliptic Poisson equation. The three methods are shown to possess the same $H^1$-stability and convergence. Some numerical tests are made, to compare the numerical results from the three methods and to review our theoretical results.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.030513.200513a }, url = {http://global-sci.org/intro/article_detail/eajam/10853.html} }
TY - JOUR T1 - $H^1$-Stability and Convergence of the FE, FV and FD Methods for an Elliptic Equation AU - Yinnian He & Xinlong Feng JO - East Asian Journal on Applied Mathematics VL - 2 SP - 154 EP - 170 PY - 2018 DA - 2018/02 SN - 3 DO - http://doi.org/10.4208/eajam.030513.200513a UR - https://global-sci.org/intro/article_detail/eajam/10853.html KW - Finite element method, finite difference method, finite volume method, Poisson equation, stability and convergence. AB -

We obtain the coefficient matrices of the finite element (FE), finite volume (FV) and finite difference (FD) methods based on $P_1$-conforming elements on a quasi-uniform mesh, in order to approximately solve a boundary value problem involving the elliptic Poisson equation. The three methods are shown to possess the same $H^1$-stability and convergence. Some numerical tests are made, to compare the numerical results from the three methods and to review our theoretical results.

Yinnian He and Xinlong Feng. (2018). $H^1$-Stability and Convergence of the FE, FV and FD Methods for an Elliptic Equation. East Asian Journal on Applied Mathematics. 3 (2). 154-170. doi:10.4208/eajam.030513.200513a
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