AUTHORS: Yuri K. Dem’yanovich

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The smoothness of functions is absolutely essential in the case of space of functions in the finit element method (FEM): incompatible FEM slowly converges and has evaluations in nonstandard metrics. The interest in smooth approximate spaces is supported by the desire to have a coincidence of smoothness of an exact solution and an approximate one. The construction of smooth approximating spaces is the main problem of the finit element method. A lot of papers have been devoted to this problem. The embedding of FEM spaces is another important problem; the last one is extremely essential in different approaches to approximate problems, speeding up of convergence and wavelet decomposition. This paper is devoted to coordinate functions obtained with approximate relations which are a generalization of the Strang-Michlin’s identities. The aim of this paper is to discuss the pseudo-smoothness of mentioned functions and embedding of relevant FEM spaces. Here we have the necessary and sufficien conditions for the pseudo-smoothness, definitio of maximal pseudo-smoothness and conditions of the embedding for FEM spaces define on embedded subdivisions of smooth manifold. The relations mentioned above concern the cell decomposition of differentiable manifold. The smoothness of coordinate functions inside the cells coincides with the smoothness of the generating vector function of the right side of approximate relations so that the main question is the smoothness of the transition through the boundary of the adjacent cells. The smoothness in this case is the equality of values of functionals with supports in the adjacent cells. The obtained results give the opportunity to verify the smoothness on the boundary of support of basic functions and after that to assert that basic functions are smooth on the whole. In conclusion it is possible to say that this paper discusses the smoothness as the general case of equality of linear functionals with supports in adjacent cells of differentiable manifold. The results may be applied to different sorts of smoothness, for example, to mean smoothness and to weight smoothness. They can be used in different investigations of the approximate properties of FEM spaces, in multigrid methods and in the developing of wavelet decomposition

KEYWORDS: finit element method, general smoothness, embedded spaces, minimal splines, approximation on manifold

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