On Simplex Trapezoidal Rule Families

We consider quadrature rules of a special type for the n-dimensional simplex. These form natural generalizations of the one-dimensional trapezoidal rule, and employ function values at points on a symmetric equally spaced grid, which includes the simplex vertices. The weight assigned to an abscissa is proportional to $m^{ - n} $, is the same at each interior grid point, and is the same at each grid point lying within the same d-dimensional boundary simplex.Families of such quadrature rules are defined, and some of their properties are derived. The principal result is the derivation of an error expansion for each rule; this expansion is an asymptotic expansion in inverse powers of m for the quadrature discretization error. Conditions under which this expansion is even are obtained together with the conditions under which the rule family is symmetric, or is of polynomial degree zero or one.