Order/Degree of accuracy for multidimensional integration using Gauss-Legendre quadrature

Question

Let we have continuous function $f(x,y)$ in all $(x,y)\in(-\infty,\infty)$, and we would take the finite integration of $f$: $$\int_{a}^{b}{\int_{p}^{q}{f(x,y) \,dx \,dy}}$$ and we take the integration by using Gaussian quadrature, e.g. $n$ points quadrature in each directions: $$\int_{a}^{b}{\int_{p}^{q}{f(x,y) \,dx \,dy}} \simeq {\frac{(b-a)(q-p)}{4} \sum_{i=1}^{n}{\sum_{j=1}^{n}{w_i w_j f\left({\frac{b-a}{2} \xi_i + \frac{a+b}{2}},\frac{q-p}{2}\xi_j+\frac{p+q}{2}\right)}}}.$$ Of course the integration have a numerical error, for one dimensional integration, Gauss-Legendre quadrature would have degree $2n-1$, what degree of accuracy of $n$ points quadrature in this multidimensional case?

Answer 1

I hope this still helps you: The 2-d quadrature rule you describe can be constructed by first using the 1-d rule in x-direction, then in y-direction:

Assume $f$ to be sufficiently smooth. Then we have $$ \int_a^b \int_p^q f(x,y)\,dx\,dy = (q-p)\int_a^b \sum_{i=1}^n\omega_if\left(x,\frac{q-p}2\xi_i+\frac{q+p}2\right)+\mathcal O\left((q-p)^{2n}\right)\,dy\\ = (q-p)(b-a)\sum_{j=1}^n \sum_{i=1}^n\omega_i\omega_jf\left(\frac{b-a}2\xi_i+\frac{a+b}2,\frac{q-p}2\xi_i+\frac{q+p}2\right)+\mathcal O\left((b-a)^{2n}\right) + \mathcal O\left((q-p)^{2n}\right). $$ This means that you still have the same order of accuracy.