Quadrature Formula, FEM-Approximation (in Russian)

Abstract

The author has constructed in one of his preceeding papers a FEM-approximation formula for functions of the class $C^{(s)}[0,1]$ ($s$ an arbitrary natural number) which depends only on the values of the given function at the knots of the net. This construction demands to continue the given function (with preserving the class $C^{(s)}$ on a little segment of one or both sides of the segment [0, 1]. Accordingly one obtains two approximate formulae. Integrating them one obtains two quadrature formulae which accuracy depends from $s$. The right-hand side of every of them is the sum of the right-hand side of the classical trapezoid rule and some boundary layer. If one use bilateral continuation, then the boundary layer becomes a little more complicated, but the accuracy of the quadrature formula increases.