Symmetric quadrature rules on a triangle. (English) Zbl 1050.65022
Summary: We present a class of quadrature rules on triangles in \(\mathbb R^2\) which, somewhat similar to Gaussian rules on intervals in \(\mathbb R^1\), have rapid convergence, positive weights, and symmetry. By a scheme combining simple group theory and numerical optimization, we obtain quadrature rules of this kind up to the order 30 on triangles. This scheme, essentially a formalization and generalization of the approach used by J. N. Lyness and D. Jespersen [J. Inst. Math. Appl. 15, 19–32 (1975; Zbl 0297.65018)] over 25 years ago, can be easily extended to other regions in \(\mathbb R^2\) and surfaces in higher dimensions, such as squares, spheres. We present example formulae and relevant numerical results.
MSC:
| 65D32 | Numerical quadrature and cubature formulas |
| 41A55 | Approximate quadratures |
| 90C59 | Approximation methods and heuristics in mathematical programming |
Keywords:
Symmetric quadrature; Triangle; Gaussian quadrature; convergence; positive weights; symmetry; numerical resultsCitations:
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