Quadrature over a Simplex: Part 2. A Representation for the Error Functional

A particular subdivision of a simplex Sₙ into mⁿ subsimplices of equal n-volume is described. The mⁿ-copy quadrature rule Q⁽ᵐ⁾ (Sₙ ) is defined in terms of a set \ Q\ of n! simplex quadrature rules, one for each distinct type of subsimplex occurring in the subdivision. An n-dimensional analogue of the Euler–Maclaurin asymptotic expansion for the simplex valid for analytic integrand functions φ and any quadrature rule set \ Q\ is derived by integrating the representation for φ ( x) obtained in Part 1. This has the form\[{{Q⁽ᵐ⁾ (S_n )φ - I(S_n )φ Σ A_q (Q;φ )} / {m^q }}\] and terminates when φ ( x) is a polynomial. For rule sets based on many familiar simplex rules, the odd terms vanish leaving an even expansion