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J. Comp. Math., 24 (2006), pp. 281-294. |
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Implementation Of Mixed Methods As Finite Difference Methods And Applications To Nonisothermal Multiphase Flow In Porous Media Zhang-xin Chen 1, Xi-jun Yu 2 1 Center for Scientific Computation, Box 750156, Southern Methodist University, Dallas, TX 75275-0156, USA/Research Center for Science, Xi'an Jiaotong University, Xi'an 710049, China/Center for Advanced Reservoir Modeling and Simulation, College of Engineering, Peking University, China2 Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China Received 2006-3-1 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.
Key words: Finite difference; Implementation; Mixed method; Error estimates; Superconvergence; Tensor coefficient; Nonisothermal multiphase; Multicomponent flow; Porous media. |