Trapezoidal rule for multiple integrals over hyperquadrilaterals (Q1126610)

The author obtains a trapezoidal quadrature formula for multiple integrals which is applicable to nonsymmetric lattices. The domain of integration is an \(N\)-dimensional hyperquadrilateral. This is partitioned into a set of \(N\)-dimensional hypertriangles. In each of these \(2^NN!\) hypertriangles, the integral of a linear function of the \(N\) coordinates is simply the arithmetic average of the values of the integrand at the \(N+1\) apices of the hypertriangle times the hyperarea of the hypertriangle. A multiple integral over the hyperarea of the hypertriangle. A multiple integral over the hyperquadrilateral is well approximated by the integral of a piecewise linear function that takes on the values of the integrand at \(3^N\) nodal points of the hyperquadrilateral, one of which is an interior point and \(2^N\) of which are vertical points. The approximating integral is equal to a sum of \(2^NN!\) terms, each being an arithmetic average times the hyperarea of a hypertriangle. The result is a weighted sum of the \(3^N\) nodal values. The obtained trapezoidal rule is applied to a surface integral to obtain finite-sum expressions for partial derivatives of a function of three variables in non-orthogonal coordinates.