A monotone finite element method for reaction-drift-diffusion equations with discontinuous reaction coefficients
<|reference_start|>A monotone finite element method for reaction-drift-diffusion equations with discontinuous reaction coefficients: 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.<|reference_end|>
@article{heldman2024a,
    title={A monotone finite element method for reaction-drift-diffusion equations
  with discontinuous reaction coefficients},
    author={Max Heldman},
    journal={arXiv preprint arXiv:2407.09660},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.09660},
    primaryClass={math.NA cs.NA}
}