Generalized Gaussian quadrature rules over regions with parabolic edges

Abstract

This paper presents a generalized Gaussian quadrature method for numerical integration over regions with parabolic edges. Any region represented by R ₁={(x, y)| a≤x≤b, f(x)≤y≤g(x)} or R ₂={(x, y)| a≤y≤b, f(y)≤x≤g(y)}, where f(x), g(x), f(y) and g(y) are quadratic functions, is a region bounded by two parabolic arcs or a triangular or a rectangular region with two parabolic edges. Using transformation of variables, a general formula for integration over the above-mentioned regions is provided. A numerical method is also illustrated to show how to apply this formula for other regions with more number of linear and parabolic sides. The method can be used to integrate a wide class of functions including smooth functions and functions with end-point singularities, over any two-dimensional region, bounded by linear and parabolic edges. Finally, the computational efficiency of the derived formulae is demonstrated through several numerical examples.

Keywords:

• finite-element method
• numerical integration
• quadrature rules
• parabolic edges

2010 AMS Subject Classification:

• 65D30

Acknowledgements

The continuous research support and encouragement of the management of Amrita Vishwa Vidyapeetham is gratefully acknowledged. The authors also wish to thank the learned referee, for his/her valuable suggestions and comments which enhanced the presentation of this paper.