Smoothness of Spaces in Finite Element Methods

The smoothness of functions is absolutely essential in the case of space of functions in finite element method (FEM): incompatible FEM slowly converges and has evaluations in nonstandard metrics. Interest in smooth approximate spaces is supported by the desire to have a coincidence of smoothness of exact solution and approximate one. The construction of smooth approximating spaces is the main problem of the finite element method. A lot of papers have been devoted to this problem. The aim of the paper is the obtaining of the necessary and sufficient conditions for the smoothness of coordinate functions provided that the last ones are received by approximate relations which are a generalization of Strang-Michlin's conditions. The relations mentioned above discussed on cell decomposition of differentiable manifold. The smoothness of coordinate functions inside of cells coincides with the smoothness of generating vector function of the right side of approximate relations so that the main question is the smoothness of transition through the boundary of adjacent cells. The smoothness in this case is the equality of values of functionals with supports in the adjacent cells. The obtained results give 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 result may be applied to different sorts of smoothness, for example, to mean smoothness and to weight smoothness.