# American Institute of Mathematical Sciences

December  2006, 1(4): 689-706. doi: 10.3934/nhm.2006.1.689

## On the control volume finite element methods and their applications to multiphase flow

 1 Department of Mathematics, Box 750156, Southern Methodist University, Dallas, TX 75275-0156

Received  September 2006 Published  October 2006

In this paper we systematically review the control volume finite element (CVFE) methods for numerical solutions of second-order partial differential equations. Their relationships to the finite difference and standard (Galerkin) finite element methods are considered. Through their relationship to the finite differences, upstream weighted CVFE methods and the conditions on positive transmissibilities (positive flux linkages) are studied. Through their relationship to the standard finite elements, error estimates for the CVFE are obtained. These estimates are comparable to those for the standard finite element methods using piecewise linear elements. Finally, an application to multiphase flows in porous media is presented.
Citation: Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks & Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689
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