Hi, Karakus, I think that the Simpson method is better than Gauss Quadrature for numerical integration.

Thank you for your comment, Belhamiti. I know Simpson's rule is more accurate than trapezoidal method but not sure if it can produces more precise results than Gauss.

Gauss quadrature is optimal for polynomials of degree 2n-1. However, there may be better choices depending on the problem. If nesting property is important - then (especially for non-polynomial integrands) Clenshaw–Curtis or Gauss–Kronrod quadratures may be "better" [suited].

Here http://www.lncc.br/~alm/public/integexp.pdf authors argue that Newton-Cotes quadrature is "better" because it requires only re-computation of weights for multi-scale FE computations.

As Professor @Cara wrote, the Gaussian quadrature method is optimal for  all polynomials of degree less than or equal 2n-1, while the Newton-Cotes formulas are optimal for all polynomials of degree less than or equal n; Trapezoidal and Simpson rules are special cases of the Newton-Cotes formulas with n=1, 2. I am personally is satisfied with the Gaussian Quadrature formula. 

If you have an initial interval [a,b] and divide it in J subintervals then a composite Simpron rule requires 2J+1 function evaluations and converges with order 4. The corresponding Gauss-Legendre method requires 2J function evaluation (with 2 nodes in each subinterval) and converges with order 4. However, the constant is the error estimate of  the Simpson method is smaller than the constant in the error estimate of the GL method with 2 nodes. That may explains why Simpson rule competes the GL method with 2 nodes. As Mihai Cara wrote the GL is the only numerical integration method that is able to integrate exactly polynomials of order 2q-1 with q nodes.  If we have to solve adaptivity problems, or numerical integrations problems in higher dimensions then we have to  take into account not only the formal accuracy of the method but also the corresponding required complexity. 

In what sense other methods should be better?

Regarding precision, the Gauss quadrature is optimal. We have that for a continuous function f on the interval [a,b], the error bound | I(f)-I_n(f) |, where I_n(f) is the n^th Gaussian quadrature, can be estimated from above by 2(b-a) times the distance of f from the subspace of polynomials of degree

Hi Roman, thank you for your suggestion. Both accuracy and convergence speed are important. So, I will look into Romberg's integration as well.

It is a very difficult to say which one is better. The method depend on the nature of function and the limits used. If the function is smooth, all methods are good. If the function is too much oscillatory, then more accurate method is necessary.  

For my atomic collision physics calculations, I generally use Gauss-Legendre quadratures, sometimes Gauss-Laguerre quadratures.

1- If you have the function values at equally spaced nodes and it is not possible to obtain the function values at other nodes, then you have to use Newton-Cotes methods such as trapezoidal, simpson,... methods and to increase accuracy, Romberg's approach will help you.

2- If you can obtain the function values at any point, then the best approach is GQ methods such as Gauss-Legendre method. But in using this approach you should be careful! GQ points are all irrational points(except probably zero) and you should evaluate these point with as much as possible digits( I suggest 16 digits).

3-By increasing Gaussian points one requires increasing accuracy. But, depending on the integrand, cancellation errors will dominate the accuracy. It is necessary to obtain the optimal vale of the number of Gaussian points(my experience says that it is in the range of 7-15).

4- In case of using GQ approaches, if the length of the under consideration interval is very big, then divide it into 2, 3 or more intervals.

Why not an adaptive quadrature?

Many good answers here that refer to the numerical integration in general. However, I wonder if the author refers to integration methods as used in FE. I do not think adaptive quadratures are used in FE directly (but multi-scale FE and adaptive meshes are used which indirectly affect quality of integration). For piece-wise polynomials Gauss quadratures are perfect but overall result will depend on mesh.

Once again, thank you very much for all your useful comments and suggestions. Mihai, the integration method used in FE is Gauss-Legendre quadrature.

The equal spaced Newton-Cotes formulae (TR, Sim 1/3, 3/8 etc) are world wide used tools for numerical integrations but not used directly for singular cases. The unequal spaced symmetric and endpoint less Gauss's quadrature are applicable for both singular and non-singular cases. In some recent articles (Our research) we are trying to represent a new platform for unequal space but non symmetric endpoint less Gauss type formulae which are nicely evaluating ( with Romberg scheme) singular and non singular integrals. I think it may a reliable platform in the field of Numerical Integrations.

Yes, there are several other numerical integration methods that can be used for finite element analysis (FEA) which have their own advantages and disadvantages compared to GQ. Here are a few examples: