A method of construction of the local approximations (in particurlar – generalization of finite-element ones, for example, plane finite-elements of Courant, Zlamal, Argyris etc.) in the case of functions defined on $n$-dimensional ($n\geq1$) smooth manifold with boundary is proposed. A notion of nondegenerate simplicial subdivision of mentioned manifold is introduced, evaluations of approach and stability in Sobolev's spaces are discussed (last ones are optimal as to $N$-width of corresponding compact).