00-28

Stability of the residual free bubble method for bilinear finite elements on rectangular grids

by Franca, Leopoldo P.; Tobiska, Lutz

 

Preprint series: 00-28, Preprints

MSC:
65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
65N12 Stability and convergence of numerical methods

 

Abstract: We consider the nature of the stabilizing term arising in the residual free bubble approach for piecewise bilinear functions on rectangular grids. We show, that on the subspace of piecewise linear functions the stabilizing term is identical to that in the streamline diffusion approach. However, on the space of piecewise bilinear functions there is a case in which the stabilizing term is weaker compared to the term used in the streamline diffusion method. In the particular case when the direction of the convection is directed parallel to a diagonal of the quadrilateral, control is lost over the mixed derivatives in the convection-dominated limit.

Keywords: finite element method, convection diffusion problem, residual free bubble approach


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