Question

The extension of Gaussian quadrature to two-dimensional integrals of the form of _____

(a) I≈\(\sum_{i=1}^{n}\sum_{j=1}^{n}\)wiwjf(ξi,ηj)

(b) Natural co-ordinates

(c) w1f(ξ1)+w2f(ξ2)

(d) w1f(ξ1)

I got this question in exam.

Question is taken from Numerical Integration topic in division Two Dimensional Isoparametric Elements and Numerical Integration of Finite Element Method

Answer 1

The correct answer is (a) I≈\(\sum_{i=1}^{n}\sum_{j=1}^{n}\)wiwjf(ξi,ηj)

Best explanation: In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. An n-point Gaussian quadrature rule, is a quadrature rule constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the points xi and weights wi for i=1,…, n. The domain of integration for such a rule is conventionally taken as [−1, 1].