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A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation

Authors: Song Wang and Zi-Cai Li
Journal: Math. Comp. 72 (2003), 1689-1709
MSC (2000): Primary 65N30; Secondary 76M10
Published electronically: May 21, 2003
MathSciNet review: 1986800
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Abstract: In this paper we formulate and analyze a discretization method for a 2D linear singularly perturbed convection-diffusion problem with a singular perturbation parameter $\varepsilon$. The method is based on a nonconforming combination of the conventional Galerkin piecewise linear triangular finite element method and an exponentially fitted finite volume method, and on a mixture of triangular and rectangular elements. It is shown that the method is stable with respect to a semi-discrete energy norm and the approximation error in the semi-discrete energy norm is bounded by $\displaystyle C h\sqrt{\left \vert \frac{\ln \varepsilon}{\ln h}\right \vert} $with $C$ independent of the mesh parameter $h$, the diffusion coefficient $\varepsilon$ and the exact solution of the problem.

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Additional Information

Song Wang
Affiliation: Department of Mathematics & Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia

Zi-Cai Li
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 80424

Received by editor(s): June 7, 2001
Received by editor(s) in revised form: December 28, 2001
Published electronically: May 21, 2003
Article copyright: © Copyright 2003 American Mathematical Society