Creating and using a quadrature rule

This demo shows how quadrature rules can be obtained from Basix and how these can be used to compute the integrals of functions. A quadrature rule uses a set of points (\(p_0\) to \(p_{n-1}\)) and weights (\(w_0\) to \(w_{n-1}\)), and approximates an integral as the weighted sum of the values of the function at these points, ie

\[\int f\,\mathrm{d}x \approx \sum_i w_if(p_i).\]

First, we import Basix and Numpy.

import numpy as np

import basix
from basix import CellType, ElementFamily, LagrangeVariant

To get a quadrature rule on a triangle, we use the function make_quadrature. This function takes two or three three inputs. We want to use the default quadrature, pass in two inputs: a cell type and an order. The order of the rule is equal to the degree of the highest degree polynomial that will be exactly integrated by this rule. In this example, we use an order 4 rule, so all quartic polynomials will be integrated exactly.

make_quadrature returns two values: the points and the weights of the quadrature rule.

points, weights = basix.make_quadrature(CellType.triangle, 4)

If we want to control the type of quadrature used, we can pass in three inputs to make_quadrautre. For example, the following code would force basix to use a Gauss-Jacobi quadrature rule:

points, weights = basix.make_quadrature(
    CellType.triangle, 4, rule=basix.QuadratureType.gauss_jacobi)

We now use this quadrature rule to integrate the functions \(f(x,y)=x^3y\) and \(g(x,y)=x^3y^2\) over the triangle. The exact values of these integrals are 1/120 (0.00833333333333…) and 1/420 (0.00238095238095…) respectively.

As \(f\) is a degree 4 polynomial, we expect our quadrature rule to be able to compute its integral exactly (within machine precision). \(g\) on the other hand is a degree 5 polynomial, so its integral will not be computed exactly.

We define Python functions that compute \(f\) and \(g\) for every point. These functions use features of Numpy to compute all the values at once.

def f(points):
    return points[:, 0] ** 3 * points[:, 1]


def g(points):
    return points[:, 0] ** 3 * points[:, 1] ** 2

We can now use Numpy features to compute the integrals.

print(np.sum(weights * f(points)))
print(np.sum(weights * g(points)))

We obtain the values 0.00833333333333334 and 0.002393509368731209. As expected, the integral of \(f\) has been computed to within machine precision, while the integral of \(g\) is correct to 4 decimal places, but is not exact.

Integrating a basis function

We next use the quadrature rule to compute the integral of a basis function in a degree 3 Lagrange space. We first create the space and tabulate its basis functions at the quadrature points.

lagrange = basix.create_element(
    ElementFamily.P, CellType.triangle, 3, LagrangeVariant.equispaced)

values = lagrange.tabulate(0, points)

We compute the integral of the third (note that the counting starts at 0) basis function in this space. We can obtain the values of this basis function from values by using the indices [0, :, 3, 0]. The integral can therefore computed as follows. As this basis function will be degree three, the result will again be exact (within machine precision).

print(np.sum(weights * values[0, :, 3, 0]))