Summary
A uniform framework for the study of upwinding schemes is developed. The standard finite element Galerkin discretization is chosen as the reference discretization, and differences between other discretization schemes and the reference are written as artificial diffusion terms. These artificial diffusion terms are spanned by a four dimensional space of element diffusion matrices. Three basis matrices are symmetric, rank one diffusion operators associated with the edges of the triangle; the fourth basis matrix is skew symmetric and is associated with a rotation by ϕ/2. While finite volume discretizations may be written as upwinded Galerkin methods, the converse does not appear to be true. Our approach is used to examine several upwinding schemes, including the streamline diffusion method, the box method, the Scharfetter-Gummel discretization, and a divergence-free scheme.
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The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440
The work of this author was supported through KWF-Landis/Gyr Grant 1496, AT & T Bell Laboratories, and Cray Research
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Bank, R.E., Bürgler, J.F., Fichtner, W. et al. Some upwinding techniques for finite element approximations of convection-diffusion equations. Numer. Math. 58, 185–202 (1990). https://doi.org/10.1007/BF01385618
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DOI: https://doi.org/10.1007/BF01385618