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The Babuška-Brezzi condition and the patch test: An example. (English) Zbl 0883.65069

This paper discusses an example of a mixed finite element method in one dimension which satisfies the engineering patch test, but does not satisfy a uniform inf-sup condition. The inf-sup constant for this method is shown to be of order \(h^2\). The reasons for good results from benchmark problems for this method are addressed by characterizing the nature of input data for which the method performs well. Numerical results illustrating the effect of the non-uniform inf-sup condition are presented.

MSC:

65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
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References:

[1] Taylor, R. L.; Zienkiewicz, O. C.; Simo, J. C.; Chan, A., The patch test—a condition for assessing FEM convergence, IJNME, 22, 39-62 (1986) · Zbl 0593.73072
[2] Zienkiewicz, O. C.; Qu, S.; Taylor, R. L.; Nakazawa, S., The patch test for mixed formulations, IJNME, 23, 1873-1883 (1986) · Zbl 0614.65115
[3] Zienkiewicz, O. C.; Taylor, R. L., The patch test—necessary and sufficient conditions for convergence, (TICAM Summary Notes (1993), University of Texas) · Zbl 0918.73134
[4] Aziz, A. K.; Babuška, I., The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations (1972), Academic Press: Academic Press New York
[5] Arnold, D. N.; Babuška, I.; Osborn, J., Finite element methods: principles for their selection, Comput. Methods Appl. Mech. Engrg., 45, 57-96 (1984) · Zbl 0513.73081
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