Approximation on Manifold

Abstract

The purpose of this work is to obtain an effective evaluation of the speed of convergence for multidimensional approximations of the functions define on the differential manifold. Two approaches to approximation of functions, which are given on the manifold, are considered. The firs approach is the direct use of the approximation relations for the discussed manifold. The second approach is related to using the atlas of the manifold to utilise a well-designed approximation apparatus on the plane (finit element approximation, etc.). The firs approach is characterized by the independent construction and direct solution of the approximation relations. In this case the approximation relations are considered as a system of linear algebraic equations (with respect to the unknowns basic functions ωj (ζ)). This approach is called direct approximation construction. In the second approach, an approximation on a manifold is induced by the approximations in tangent spaces, for example, the Courant or the Zlamal or the Argyris fla approximations. Here we discuss the Courant fla approximations. In complex cases (in the multidimensional case or for increased requirements of smoothness) the second approach is more convenient. Both approaches require no processes cutting the manifold into a finit number of parts and then gluing the approximations obtained on each of the mentioned parts. This paper contains two examples of Courant type approximations. These approximations illustrate the both approaches mentioned above.