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A nodally bound-preserving discontinuous Galerkin method for the drift-diffusion equation

Gabriel R. Barrenechea, Tristan Pryer, Alex Trenam

TL;DR

This work develops two discontinuous Galerkin discretisations for the drift-diffusion (Poisson–Nernst–Planck) model: a classical interior-penalty dG method and a novel nodally bound-preserving variant that enforces non-negativity at nodal points via a convex subset variational inequality. The analyses establish well-posedness, energy dissipation, and optimal convergence in the energy norm, with the bound-preserving scheme guaranteeing a discrete maximum principle at nodes. Numerical experiments confirm both convergence rates and robust structure preservation across discontinuous data, time-dependent boundaries, and a coupled PNP system. The results provide a practically applicable framework for accurate, stable charge transport simulations that respect physical positivity constraints.

Abstract

In this work, we introduce and analyse discontinuous Galerkin (dG) methods for the drift-diffusion model. We explore two dG formulations: a classical interior penalty approach and a nodally bound-preserving method. Whilst the interior penalty method demonstrates well-posedness and convergence, it fails to guarantee non-negativity of the solution. To address this deficit, which is often important to ensure in applications, we employ a positivity-preserving method based on a convex subset formulation, ensuring the non-negativity of the solution at the Lagrange nodes. We validate our findings by summarising extensive numerical experiments, highlighting the novelty and effectiveness of our approach in handling the complexities of charge carrier transport.

A nodally bound-preserving discontinuous Galerkin method for the drift-diffusion equation

TL;DR

This work develops two discontinuous Galerkin discretisations for the drift-diffusion (Poisson–Nernst–Planck) model: a classical interior-penalty dG method and a novel nodally bound-preserving variant that enforces non-negativity at nodal points via a convex subset variational inequality. The analyses establish well-posedness, energy dissipation, and optimal convergence in the energy norm, with the bound-preserving scheme guaranteeing a discrete maximum principle at nodes. Numerical experiments confirm both convergence rates and robust structure preservation across discontinuous data, time-dependent boundaries, and a coupled PNP system. The results provide a practically applicable framework for accurate, stable charge transport simulations that respect physical positivity constraints.

Abstract

In this work, we introduce and analyse discontinuous Galerkin (dG) methods for the drift-diffusion model. We explore two dG formulations: a classical interior penalty approach and a nodally bound-preserving method. Whilst the interior penalty method demonstrates well-posedness and convergence, it fails to guarantee non-negativity of the solution. To address this deficit, which is often important to ensure in applications, we employ a positivity-preserving method based on a convex subset formulation, ensuring the non-negativity of the solution at the Lagrange nodes. We validate our findings by summarising extensive numerical experiments, highlighting the novelty and effectiveness of our approach in handling the complexities of charge carrier transport.
Paper Structure (14 sections, 21 theorems, 77 equations, 8 figures)

This paper contains 14 sections, 21 theorems, 77 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The drift-diffusion equation
  3. A temporal semi-discretisation
  4. A discontinuous Galerkin method
  5. A priori error analysis
  6. A bound-preserving method
  7. Higher-order time discretisations
  8. A priori error analysis
  9. Numerical Experiments
  10. Convergence on a uniform mesh with a smooth solution
  11. Structure preservation with discontinuous initial conditions
  12. Structure preservation with time-dependent boundary conditions
  13. An application to the coupled Poisson-Nernst-Planck system
  14. Concluding remarks

Key Result

Lemma 2.2

Let $\psi\in W^{2, \infty}\!\left( \Omega\right)$ and $0 \leq u_0 \in L^{\infty}\!\left( \Omega \right)$. If $\Delta\psi \leq 0$, then there exists a unique solution $u(t) \in L^2 \!\left( \left(0, T\right]; H^1_0\!\left( \Omega \right)\right) \cap H^1 \!\left( \left(0, T\right]; H^{-1}\!\left( \Ome

Figures (8)

  • Figure 1: Examples of piecewise polynomial functions, on a non-uniform one-dimensional mesh, which are nodally non-negative. The dotted lines denote the locations of the Lagrange nodes of the elements. For piecewise linear functions the non-negativity is global, but for higher-order polynomials the function may go negative between the nodes.
  • Figure 2: Convergence of the solution to (\ref{['eq:drift-diffusion-full-disc']}) on a uniform mesh in the $\left|\!\left|\!\left| \cdot \right|\!\right|\!\right|$-norm using piecewise polynomials of degree $p = 1, 2$. Convergence is observed at a rate of $\mathcal{O}(h^{p+1})$.
  • Figure 3: Snapshots of the solution to the example from Section \ref{['ex:drift-diffusion-structure-preservation-initial-conditions']}.
  • Figure 4: Plots comparing the nodally bound-preserving and unconstrained solutions for the example in Section \ref{['ex:drift-diffusion-structure-preservation-initial-conditions']}.
  • Figure 5: Snapshots at final time of the nodally non-negative solutions from the example in Section \ref{['ex:drift-diffusion-structure-preservation']} and comparisons to the corresponding solutions to (\ref{['eq:dg-disc']}).
  • ...and 3 more figures

Theorems & Definitions (40)

  • Remark 2.1
  • Lemma 2.2: PDE well-posedness
  • Lemma 2.3: Energy identity
  • proof
  • Lemma 2.4: Stability
  • proof
  • Lemma 2.5: Parabolic Maximum Principle Renardy2004
  • Lemma 3.1: Semi-discrete coercivity
  • proof
  • Lemma 3.2: Boundedness of $\mathscr A\xspace(\cdot, \cdot)$
  • ...and 30 more