Approximation on Manifold
2021
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.
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