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Volume 12, Issue 3
Convergence of a Cell-Centered Finite Volume Method and Application to Elliptic Equations

Gung-Min Gie & Roger Temam

Int. J. Numer. Anal. Mod., 12 (2015), pp. 536-566.

Published online: 2015-12

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  • Abstract

We study the consistency and convergence of the cell-centered Finite Volume (FV) external approximation of $H^1_0(\Omega)$, where a 2D polygonal domain $\Omega$ is discretized by a mesh of convex quadrilaterals. The discrete FV derivatives are defined by using the so-called Taylor Series Expansion Scheme (TSES). By introducing the Finite Difference (FD) space associated with the FV space, and comparing the FV and FD spaces, we prove the convergence of the FV external approximation by using the consistency and convergence of the FD method. As an application, we construct the discrete FV approximation of some typical elliptic equations, and show the convergence of the discrete FV approximations to the exact solutions.

  • Keywords

Finite Volume method, Taylor Series Expansion Scheme (TSES), convergence and stability, convex quadrilateral meshes.

  • AMS Subject Headings

65N08, 65N12, 76M12, 65N06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-12-536, author = {Gung-Min Gie and Roger Temam}, title = {Convergence of a Cell-Centered Finite Volume Method and Application to Elliptic Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2015}, volume = {12}, number = {3}, pages = {536--566}, abstract = {

We study the consistency and convergence of the cell-centered Finite Volume (FV) external approximation of $H^1_0(\Omega)$, where a 2D polygonal domain $\Omega$ is discretized by a mesh of convex quadrilaterals. The discrete FV derivatives are defined by using the so-called Taylor Series Expansion Scheme (TSES). By introducing the Finite Difference (FD) space associated with the FV space, and comparing the FV and FD spaces, we prove the convergence of the FV external approximation by using the consistency and convergence of the FD method. As an application, we construct the discrete FV approximation of some typical elliptic equations, and show the convergence of the discrete FV approximations to the exact solutions.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/501.html} }
TY - JOUR T1 - Convergence of a Cell-Centered Finite Volume Method and Application to Elliptic Equations AU - Gung-Min Gie & Roger Temam JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 536 EP - 566 PY - 2015 DA - 2015/12 SN - 12 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/501.html KW - Finite Volume method, Taylor Series Expansion Scheme (TSES), convergence and stability, convex quadrilateral meshes. AB -

We study the consistency and convergence of the cell-centered Finite Volume (FV) external approximation of $H^1_0(\Omega)$, where a 2D polygonal domain $\Omega$ is discretized by a mesh of convex quadrilaterals. The discrete FV derivatives are defined by using the so-called Taylor Series Expansion Scheme (TSES). By introducing the Finite Difference (FD) space associated with the FV space, and comparing the FV and FD spaces, we prove the convergence of the FV external approximation by using the consistency and convergence of the FD method. As an application, we construct the discrete FV approximation of some typical elliptic equations, and show the convergence of the discrete FV approximations to the exact solutions.

Gung-Min Gie and Roger Temam. (2015). Convergence of a Cell-Centered Finite Volume Method and Application to Elliptic Equations. International Journal of Numerical Analysis and Modeling. 12 (3). 536-566. doi:
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