ISSN:
0749-159X
Keywords:
Mathematics and Statistics
;
Numerical Methods
Source:
Wiley InterScience Backfile Collection 1832-2000
Topics:
Mathematics
Notes:
In the mixed finite element approximation of Stokes-like problems, stability considerations normally restrict the choice of the finite element subspaces to families satisfying the so-called inf-sup condition. However, in the case of the generalized Stokes problems that one encounters when solving either the nonstationary compressible or incompressible Navier-Stokes equations by means of some operator splitting, the contribution of the time derivative is adding a term in the continuity equations which, at least theoretically, leads to a problem that is always stable. On the other hand, the numerical solution of the discrete problems built with approximations not satisfying the inf-sup condition are generally plagued with oscillations that look pretty much like the checkerboard phenomenon occurring in the classical context. To clarify this apparent contradiction, we undertook a theoretical study of the condition number of the linear problem involving the sensitive variable, which for such a saddle point problem is the dual problem. In the course of this study, we have exhibited the relationship between that condition number and the inf-sup constant, thus providing a different point of view on the results of Brezzi and Babuska. Finally, we conducted a parametric study of the behavior of that condition number with respect to the mesh size, the viscosity, and the time step. That study showed that, even in this context, the use of “stable element” was adding to the quality of the approximation by reducing the condition number to an acceptable level.
Additional Material:
8 Ill.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1002/num.1690080402
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