Some criteria for numerically integrated matrices and quadrature formulas for triangles

Abstract

SUMMARYFor a wide class of finite element matrices integrated numerically rather than exactly, a definable number of sampling points is found to be sufficient for keeping their theoretical properties unchanged. A systematic criterion limiting the number of possible point configurations for numerical quadrature formulas on triangles is established. Some new high order formulas are presented. Tables containing optimal formulas with respect to minimum number of sampling points and required degrees of accuracy are given. They are arranged so as to assist with selection of suitable quadrature formulas for finite element computer programming.

Gauss Legendre–Gauss Jacobi quadrature rules over a tetrahedral region

“…Laurie [10] derived a 7-point integration rule and discussed the numerical error in integrating some functions. Laursen and Gellert [11] gave a table of symmetric integration formulae up to a precision of degree ten. Dunavant [12] presented some extensions to the integration formulae given by Lyness and Jespersen [8] and also gave tables of integration formulae with precisions of degree from 11 to 20.…”

On the application of two Gauss–Legendre quadrature rules for composite numerical integration over a tetrahedral region

“…Laurie [10] derived a 7-point integration rule and discussed the numerical error in integrating some functions. Laursen and Gellert [11] gave a table of symmetric integration formulae up to a precision of degree ten. Dunavant [12] presented some extensions to the integration formulae given by Lyness and Jespersen [8] and also gave tables of integration formulae with precisions of degree from eleven to twenty.…”

“…Reddy [17] and Reddy and Shippy [18] derived the 3-point, 4-point, 6-point and 7-point rules of precision 3, 4, 6 and 7 respectively which gave improved accuracy. Since the precision of all the formulae derived by the authors [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] is limited to a precision of degree ten and it is not likely that the techniques can be extended much further to give a greater accuracy which may be demanded in future, Lague and Baldur [19] proposed the product formulae based only on the sampling points and weight coefficients of Gauss-Legendre quadrature rules. By the proposed method of [19] this restriction is removed and one can now obtain numerical integration rules of very high degree of precision as the derivation now rely on standard Gauss-Legendre Quadarature rules.…”

Gauss Legendre Quadrature Formulae for Tetrahedra

“…Laurie [10] derived a seven-point integration rule and discussed the numerical error in integrating some functions. Laursen and Gellert [11] gave a table of symmetric integration formulae up to a precision of degree ten. Lether and Hillion [12,13] derived the formulae for triangles as product of one-dimensional Gauss Legendre and Gauss Jacobi quadrature rules.…”

“…Reddy [15] and Reddy and Shippy [16] derived three-point, four-point, six-point, seven-point of precision 3, 4, 6 and 7 respectively, which gave improved accuracy. Since the precision of all the formulae derived by the authors [4][5][6][7][8][9][10][11][12][13][14][15][16] is limited to a precision of degree ten and it is not likely that the techniques can be extended much further to give a greater accuracy, which may be demanded in future, Lague and Baldur [17] proposed product formulae based only on the roots and weight co-efficients of Gauss Legendre quadrature rules. By the proposed method, this restriction is removed and one can now obtain numerical integration of very high degree of precision as the derivations now rely on standard Gauss Legendre quadarature rules [17].…”