Volume 24, Issue 3
Implementation of Mixed Methods as Finite Difference Methods and Applications to Nonisothermal Multiphase Flow in Porous Media

Zhang-Xin Chen & Xi-Jun Yu

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J. Comp. Math., 24 (2006), pp. 281-294.

Published online: 2006-06

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  • Abstract

In this paper we consider mixed finite element methods for second order elliptic problems. In the case of the lowest order Brezzi-Douglas-Marini elements (if $d=2$) or Brezzi-Douglas-Dur\'an-Fortin elements (if $d=3$) on rectangular parallelepipeds, we show that the mixed method system, by incorporating certain quadrature rules, can be written as a simple, cell-centered finite difference method. This leads to the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a diagonal tensor coefficient, the sparsity pattern for the scalar unknown is a five point stencil if $d=2$, and seven if $d=3$. For a general tensor coefficient, it is a nine point stencil, and nineteen, respectively. Applications of the mixed method implementation as finite differences to nonisothermal multiphase, multicomponent flow in porous media are presented.

  • Keywords

Finite difference Implementation Mixed method Error estimates Superconvergence Tensor coefficient Nonisothermal multiphase Multicomponent flow Porous media

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@Article{JCM-24-281, author = {}, title = {Implementation of Mixed Methods as Finite Difference Methods and Applications to Nonisothermal Multiphase Flow in Porous Media}, journal = {Journal of Computational Mathematics}, year = {2006}, volume = {24}, number = {3}, pages = {281--294}, abstract = { In this paper we consider mixed finite element methods for second order elliptic problems. In the case of the lowest order Brezzi-Douglas-Marini elements (if $d=2$) or Brezzi-Douglas-Dur\'an-Fortin elements (if $d=3$) on rectangular parallelepipeds, we show that the mixed method system, by incorporating certain quadrature rules, can be written as a simple, cell-centered finite difference method. This leads to the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a diagonal tensor coefficient, the sparsity pattern for the scalar unknown is a five point stencil if $d=2$, and seven if $d=3$. For a general tensor coefficient, it is a nine point stencil, and nineteen, respectively. Applications of the mixed method implementation as finite differences to nonisothermal multiphase, multicomponent flow in porous media are presented. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8752.html} }
TY - JOUR T1 - Implementation of Mixed Methods as Finite Difference Methods and Applications to Nonisothermal Multiphase Flow in Porous Media JO - Journal of Computational Mathematics VL - 3 SP - 281 EP - 294 PY - 2006 DA - 2006/06 SN - 24 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8752.html KW - Finite difference KW - Implementation KW - Mixed method KW - Error estimates KW - Superconvergence KW - Tensor coefficient KW - Nonisothermal multiphase KW - Multicomponent flow KW - Porous media AB - In this paper we consider mixed finite element methods for second order elliptic problems. In the case of the lowest order Brezzi-Douglas-Marini elements (if $d=2$) or Brezzi-Douglas-Dur\'an-Fortin elements (if $d=3$) on rectangular parallelepipeds, we show that the mixed method system, by incorporating certain quadrature rules, can be written as a simple, cell-centered finite difference method. This leads to the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a diagonal tensor coefficient, the sparsity pattern for the scalar unknown is a five point stencil if $d=2$, and seven if $d=3$. For a general tensor coefficient, it is a nine point stencil, and nineteen, respectively. Applications of the mixed method implementation as finite differences to nonisothermal multiphase, multicomponent flow in porous media are presented.
Zhang-Xin Chen & Xi-Jun Yu. (1970). Implementation of Mixed Methods as Finite Difference Methods and Applications to Nonisothermal Multiphase Flow in Porous Media. Journal of Computational Mathematics. 24 (3). 281-294. doi:
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