Qualitative Comparison Between Performance Of Quadrature Rules For Triangles And Squares

Abstract

Numerical integration on an arbitrary compact set  D  ⊂  ℝ²  generally goes through two phases. The first consists in discretizing D into simple finite elements. The second phase consists in calculating an approximation of the integral on each finite element, to deduce from it an approximation of the integral on D. The triangle and the square are the finite elements most used in affine discretization. Constructing an effective quadrature rule is a laborious and time-consuming task. The quadrature rules discovered so far owe much to the power of supercomputers. This research topic, more than a century old, is still relevant thanks to the continuous growth of supercomputers. We describe in this study the different approaches that allowed the development of positive interior symmetric (PIS) quadrature rules for the triangle and the square. Next, we determine the strength and relative error associated with each rule. Additionally, we use Genz test functions to assess the accuracy of different quadrature rules. Finally, we propose two techniques to reduce the integration error inherent in non-regular integrands.