Question 1

Why is Gauss-Legendre quadrature so accurate with few points?

Accepted Answer

Gauss-Legendre quadrature optimally chooses both the evaluation points and weights so that an n-point rule integrates polynomials of degree up to 2n−1 exactly. Unlike equally-spaced rules (trapezoidal, Simpson), the Gauss points are non-uniform and positioned at roots of Legendre polynomials. A 5-point rule is exact for degree-9 polynomials and achieves much lower errors for smooth functions with the same number of function evaluations.

Question 2

What is Romberg integration?

Accepted Answer

Romberg integration applies Richardson extrapolation repeatedly to the Trapezoidal rule results at successively halved step sizes T(h), T(h/2), T(h/4), … building a triangular table. Each extrapolation step increases the order of accuracy by 2, reaching O(h^(2k)) after k steps. For smooth integrands it achieves very high accuracy with relatively few function evaluations.

Question 3

Is Simpson's rule always more accurate than the Trapezoidal rule?

Accepted Answer

Typically yes. Simpson's 1/3 rule has error O(h⁴) versus O(h²) for the Trapezoidal rule, so for the same number of subintervals it is generally more accurate. However, for non-smooth functions (discontinuities, sharp peaks) the advantage diminishes. For very few subintervals the function shape can occasionally make the Trapezoidal rule competitive.

Question 4

How is numerical integration used in FEM (Finite Element Method)?

Accepted Answer

FEM uses numerical integration (Gauss quadrature) to compute element stiffness matrices and load vectors. Linear elements typically use 1-point Gauss integration; quadratic elements use higher-order Gauss points. Stresses are computed at Gauss points and then extrapolated to nodes for post-processing. The number and location of Gauss points directly affects both accuracy and computational cost.

Numerical Integration Methods Comparator

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