A semi-analytical scheme for highly oscillatory integrals over tetrahedra

Author (s): Hospital-Bravo, R.; Sarrate, J. and Díez, P.
Journal: International Journal for Numerical Methods in Engineering

Volume: 111, Issue 8
Pages: 703 – 723
Date: 2017

This paper details a semi-analytical procedure to efficiently integrate the product of a smooth function and a complex exponential over tetrahedral elements. These highly oscillatory integrals appear at the core of different numerical techniques. Here, the Partition of Unity Method (PUM) enriched with plane waves is used as motivation. The high computational cost or the lack of accuracy in computing these integrals is a bottleneck for their application to engineering problems of industrial interest. In this integration rule, the non-oscillatory function is expanded into a set of Lagrange polynomials. In addition, Lagrange polynomials are expressed as a linear combination of the appropriate set of monomials, whose product with the complex exponentials is analytically integrated, leading to 16 specific cases that are developed in detail. Finally, we present several numerical examples to assess the accuracy and the computational efficiency of the proposed method, compared to standard Gauss-Legendre quadratures.



@article {NME:NME5474,
author = {Hospital-Bravo, R. and Sarrate, J. and Díez, P.},
title = {A semi-analytical scheme for highly oscillatory integrals over tetrahedra},
journal = {International Journal for Numerical Methods in Engineering},
volume = {111},
number = {8},
issn = {1097-0207},
url = {http://dx.doi.org/10.1002/nme.5474},
doi = {10.1002/nme.5474},
pages = {703--723},
keywords = {highly oscillatory integral, semi-analytical integration rule, Lagrange polynomials, Helmholtz equation, partition of unity method (PUM), plane waves},
year = {2017},
note = {nme.5474},