The Hexahedron Domain

The reference domain for a hexahedral element is the unit cube $\hat{T} = [0,1]^3$ . Every hexahedral element in a mesh is the image of $\hat{T}$ under a trilinear map $F_e$ .

Just as the quadrilateral rule is a tensor product of two 1D Gauss–Legendre rules, the hexahedron rule is a tensor product of three. The derivation follows exactly the same reasoning, extended to a third coordinate direction. The $n^3$ quadrature points are all triples $(\hat{x}_i, \hat{x}_j, \hat{x}_k)$ of 1D nodes, with weights $w_i w_j w_k$ . All weights are positive, sum to $1$ , and the rule integrates all monomials $x^a y^b z^c$ with $a, b, c \leq p$ exactly.

The figure below shows the order-5 rule, which places $3 \times 3 \times 3 = 27$ points in a regular grid inside the reference cube.

Tensor-product quadrature points on the reference hexahedron, order 5

In the library, this construction is implemented in refdomains/hexahedron.py by HexahedronDomain, which delegates to LineDomain and forms the tensor product using np.meshgrid. Its quadrature(order) method supports any polynomial order for which LineDomain has a rule.