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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

A monotone finite element scheme
for convection-diffusion equations


Authors: Jinchao Xu and Ludmil Zikatanov
Journal: Math. Comp. 68 (1999), 1429-1446
MSC (1991): Primary 65N30, 65N15
Published electronically: May 20, 1999
MathSciNet review: 1654022
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Abstract | References | Similar Articles | Additional Information

Abstract: A simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection-diffusion problems in any spatial dimension by properly averaging the PDE coefficients on element edges. The resulting finite element stiffness matrix is an $M$-matrix under some mild assumption for the underlying (generally unstructured) finite element grids. As a consequence the proposed edge-averaged finite element scheme is particularly interesting for the discretization of convection dominated problems. This scheme admits a simple variational formulation, it is easy to analyze, and it is also suitable for problems with a relatively smooth flux variable. Some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems.


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Additional Information

Jinchao Xu
Affiliation: Center for Computational Mathematics and Applications, Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: xu@math.psu.edu

Ludmil Zikatanov
Affiliation: Center for Computational Mathematics and Applications, Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: ltz@math.psu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01148-5
PII: S 0025-5718(99)01148-5
Keywords: Convection dominated problems, finite element method, monotone schemes, up-winding, Scharfetter-Gummel discretization, error bounds
Received by editor(s): May 6, 1996
Received by editor(s) in revised form: December 16, 1997
Published electronically: May 20, 1999
Additional Notes: The first author’s work was partially supported by NSF DMS94-03915-1 and NSF DMS-9706949 through Penn State, and by NSF ASC-92-01266 and ONR-N00014-92-J-1890 through UCLA
The second author’s work was partially supported by the Bulgarian Ministry of Education and Science Grant I–504/95, by NSF Grant Int-95–06184 and ONR-N00014-92-J-1890 through UCLA, and also by the Center for Computational Mathematics and Applications of Pennsylvania State University.
Article copyright: © Copyright 1999 American Mathematical Society