On the efficient evaluation of certain integrals in the Galerkin F. E. Method

The use of linear finite elements in fluid dynamic problems requires the evaluation of integrals of polynomial expressions, which arise from product terms in the equations of motion. An algorithm based on Simpson quadrature is presented and its efficiency compared with that of the more usual one, based on Gaussian quadrature. For both algorithms, the integrations are exact provided that the polynomial integrand is at most cubic. It is found that the Simpson algorithm is twice as efficient as the corresponding Gaussian one, for the evaluation of integrals in one, two and three space dimensions. This doubling of efficiency is a consequence of the vanishing of the basis functions at certain points, a property that can be exploited in the Simpson algorithm, but not in the Gaussian one. It is thought that the use of the Simpson algorithm will prove to be beneficial in many finite element fluid dynamic codes, because the evaluation of product terms generally represents a significant fraction of the total computational cost.