CONSTRUCTION OF INTERPOLATING POLYNOMIAL OF THE 5TH DEGREE ON ARBITRARY TRIANGLE BY USING BASIS POLYNOMIALS ON THE “UNIT” TRIANGLE - Litvin, O. M.; Kovalenko, G. V.; Denisova, O. I. (Computer Science and Applied Mathematics, 2016)
In this article, we consider the construction of the interpolating polynomial of Zlamal-Zenisek 5th degree on an arbitrary triangle. The proposed scheme for the construction of the interpolating polynomial is based on the use of the basis polynomials on the “unit” triangle (with vertices (0,0), (1,0), (0,1)). We obtain explicit formulas for all twenty-one basis polynomials. To find these basic polynomials we assume known values of interpolated function and its partial derivatives up to second order in all the vertices of the triangle, and the values of the derivatives in the direction normal to the inner sides of the triangle, calculated in the middle of the sides. Constructed basic polynomials have certain interpolation properties that allows you to record the interpolation polynomial in the form of a linear combination of basis polynomials that their coefficients are exactly equal or function values, or the values of its partial derivatives up to second order in the vertices of a triangle, or the value of the derivative in the direction of the inward normal to the sides of the triangle, calculated in the middle of the sides. Basic polynomials on an arbitrary triangle is defined as the composite function of the basis polynomials on “unit” triangle, and linear functions, which define an affine transformation that maps an arbitrary triangle into the “unit” triangle. But coefficients in the expression for the interpolating polynomial will not be exactly equal to the values of the interpolated function and certain of its derivatives. We introduce into consideration an auxiliary operator of interpolation Sf  x, y for calculating part of these coefficients. This operator is defined as a linear combination of basis polynomials obtained on the basis of 18 basis polynomials for “unit” triangle, which stand at the values of the interpolated function and its partial derivatives. We have obtained expressions for the coefficients of the operator Sf  x, y at which it satisfies the interpolation conditions: its value and value of its partial derivatives (up to the second order) at the vertices triangle are equal to the corresponding values of the interpolated function and its derivatives. Our main result is the construction of the operator of interpolation Of  x, y , which equal to the sum of operator Sf  x, y and a linear combination of 3 basis polynomials obtained using basis polynomials for the “unit” triangle. These basis polynomials arranged at the values of the normal derivative of the interpolated function. It is shown that the operator Of  x, y has all the interpolation properties of the operator Sf  x, y regardless of the coefficients at the basis polynomials in the differenceOf  x, y  Sf  x, y . We have obtained expressions for these coefficients, at which operator has the interpolation property: value of its derivatives with respect to the direction normal to the inner sides of the triangle, calculated in the middle of the sides are equal to the corresponding derivatives of the interpolated function.