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Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems


Authors: So-Hsiang Chou, Do Y. Kwak and Kwang Y. Kim
Journal: Math. Comp. 72 (2003), 525-539
MSC (2000): Primary 65F15, 65N30, 35J60
DOI: https://doi.org/10.1090/S0025-5718-02-01426-6
Published electronically: March 21, 2002
MathSciNet review: 1954955
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Abstract: We construct and analyze a mixed finite volume method on quadrilateral grids for elliptic problems written as a system of two first order PDEs in the state variable (e.g., pressure) and its flux (e.g., Darcy velocity). An important point is that no staggered grids or covolumes are used to stabilize the system. Only a single primary grid system is adopted, and the degrees of freedom are imposed on the interfaces. The approximate flux is sought in the lowest-order Raviart-Thomas space and the pressure field in the rotated-$Q1$ nonconforming space. Furthermore, we demonstrate that the present finite volume method can be interpreted as a rotated-$Q1$ nonconforming finite element method for the pressure with a simple local recovery of flux. Numerical results are presented for a variety of problems which confirm the usefulness and effectiveness of the method.


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Additional Information

So-Hsiang Chou
Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403
Email: chou@bgnet.bgsu.edu

Do Y. Kwak
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, Taejon, Korea 305-701
Email: dykwak@math.kaist.ac.kr

Kwang Y. Kim
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, Taejon, Korea 305-701
Email: kky@mathx.kaist.ac.kr

DOI: https://doi.org/10.1090/S0025-5718-02-01426-6
Keywords: Mixed method, finite volume method, box method, quadrilateral grid
Received by editor(s): November 14, 2000
Received by editor(s) in revised form: May 29, 2001
Published electronically: March 21, 2002
Additional Notes: The research of the first author was supported by NSF grant DMS-0074259
The research of the second and third authors was supported by BK21 project, Korea and by grant No. 2000-2-10300-001-5 from the Basic Research Program of the Korea Science & Engineering Foundation
Article copyright: © Copyright 2002 American Mathematical Society

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