TRAPEZOIDAL RULE FOR MULTIPLE INTEGRALS OVER HYPERQUADRILATERALS

Trapeioidal rule for a multiple integral over a hyperquadrilateral is devised. The N-dimensional hyperquadrilateral is partitioned into 2(N) N! hypertriangles, in each of which the integrand is interpolated by a linear function of the N coordinate variables. The resulting N-dimens ional trapezoidal rule is a useful quadrature formula for approximatin g a multiple integral by a weighted sum of the values of the integrand at the nodes of a hyperquadrilateral lattice. For multiple integrals, likely the trapezoidal rule gives better approximation than higher-de gree rules when the dimensionality is high. The latter interpolatory r ules have the shortcoming that some of the basis polynomials may be no t everywhere nonnegative, incurring the possibility of rendering the a ssociated nodal weights negative. As an application, the trapezoidal r ule is applied to a surface integral to obtain finite-sum expressions for partial derivatives of a function of three variables in non-orthog onal coordinates. The obtained finite-sum approximation has better acc uracy than corresponding finite-difference approximation by accounting for the coupling effect of multiple dimensionality. (C) Elsevier Scie nce Inc., 1997.