  We prove an abstract convergence result for a family of dual-mesh based
quadrature rules on tensor products of simplical meshes. In the context of the
multilinear tensor-product finite element discretization of
reaction-drift-diffusion equations, our quadrature rule generalizes the
mass-lump rule, retaining its most useful properties; for a nonnegative
reaction coefficient, it gives an $O(h^2)$-accurate, nonnegative
diagonalization of the reaction operator. The major advantage of our scheme in
comparison with the standard mass lumping scheme is that, under mild
conditions, it produces an $O(h^2)$ consistency error even when the integrand
has a jump discontinuity. The finite-volume-type quadrature rule has been
stated in a less general form and applied to systems of reaction-diffusion
equations related to particle-based stochastic reaction-diffusion simulations
(PBSRD); in this context, the reaction operator is \textit{required} to be an
$M$-matrix and a standard model for bimolecular reactions has a discontinuous
reaction coefficient. We apply our convergence results to a finite element
discretization of scalar drift-diffusion-reaction model problem related to
PBSRD systems, and provide new numerical convergence studies confirming the
theory.
