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\(b=\int g\). (English) Zbl 0904.76041
The basic reason for introducing stabilized methods is that straightforward application of the Galerkin version of the finite element method to certain problems of mathematical physics and engineering yields numerical approximations that are deficient in that they do not inherit the stability properties of the continuous problem. In recent years several efforts have undertaken to better understand the theoretical foundations and origin of stabilized methods. Two approaches have achieved particular success. The first is based on the identification of stabilized methods with Galerkin finite element methods employing finite element spaces enriched with the so-called ‘bubble’ functions. Another approach is the variational multiscale method.
In this paper it is shown that variational multiscale and the residue-free bubbles concepts are equivalent, and consequently both of them exhibit the same attributes and shortcomings.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
74S05 Finite element methods applied to problems in solid mechanics
76R99 Diffusion and convection
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