Symmetric Quadrature Formulae for Simplexes

Abstract

Symmetric interpolation polynomials are defined for iV-dimensional simplexes with the aid of a symmetric coordinate notation.These polynomials are used to produce symmetric interpolatory quadrature formulae of arbitrary degree of precision over simplexes of arbitrary dimensionality.Tabulated values of weight coefficients are given for triangles and tetrahedra.Under this definition, the size of a 1-simplex is its length, that of a 2-simplex its area, and so on.Let a point P he located within the simplex S. Let >Sm denote the simplex defined by P and the vertices of S other than the rath, i.e., by vertices 1, 2, -, ra -1, m + 1, -, N, N -\-1 of <S, and the point P. Sm is contained entirely within S. Any interior point P thus defines a unique partitioning of S into N + 1 subsimplexes Sk, k = 1, 2, -, N 4-1.Using Eq. (1), it is readily verified that N+X (2) '(&)=*().