  We present new and efficient quadrature rules for computing the stiffness
matrices of mass-lumped tetrahedral elements for wave propagation modelling.
These quadrature rules allow for a more efficient implementation of the
mass-lumped finite element method and can handle materials that are
heterogeneous within the element without loss of the convergence rate. The
quadrature rules are designed for the specific function spaces of recently
developed mass-lumped tetrahedra, which consist of standard polynomial function
spaces enriched with higher-degree bubble functions. For the degree-2
mass-lumped tetrahedron, the most efficient quadrature rule seems to be an
existing 14-point quadrature rule, but for tetrahedra of degrees 3 and 4, we
construct new quadrature rules that require less integration points than those
currently available in the literature. Several numerical examples confirm that
this approach is more efficient than computing the stiffness matrix exactly and
that an optimal order of convergence is maintained, even when material
properties vary within the element.
