A monotone finite element method for reaction-drift-diffusion equations with discontinuous reaction coefficients

TL;DR

This paper introduces a new finite element quadrature rule for reaction-drift-diffusion equations that maintains accuracy and positivity even with discontinuous reaction coefficients, improving upon standard mass-lumping methods.

Contribution

It develops a generalized quadrature rule that achieves second-order accuracy and positivity preservation for reaction-drift-diffusion equations with discontinuous coefficients, extending previous mass-lumping techniques.

Findings

The proposed scheme achieves $O(h^2)$ accuracy with discontinuous integrands.

It produces a nonnegative diagonalization of the reaction operator.

Numerical experiments confirm the theoretical convergence results.

Abstract

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.