Prove convergence of the mass-lumped finite-element discretization

Prove that the additional consistency error introduced by mass-lumping quadrature does not change the convergence rate of the finite-element discretization of the reaction-drift-diffusion model in the $H^1$ or $L^2$ norm.

Background

The CRDME derivation applies mass-lumping quadrature to all particle coordinates except the coordinate associated with the particle being transported, for which an edge-averaged finite element discretization is used. The paper notes that mass-lumping has a formal O(h2)O(h^2) quadrature error for sufficiently regular functions, but does not establish that this additional error preserves the convergence rate of the overall finite-element approximation in the relevant spatial norms.

Resolving this problem would provide an analytical justification for the observed numerical convergence of the spatial discretization and would connect the multiparticle CRDME construction to existing estimates for mass-lumped parabolic and reaction-drift-diffusion finite-element methods.

References

Although we do not prove this here, our numerical convergence studies support our expectation that the additional consistency error introduced by mass-lumping quadrature does not change the convergence rate of our finite element discretization in either the $H1$ or $L2$ norm.

— A Convergent Reaction-Drift-Diffusion Master Equation with Interaction Potentials  (2609.09546 - Heldman et al., 9 Sep 2026) in Section 6.1, “Hopping rate derivation” (around Eq. (\ref{eq:mass-lump-quadrature}))