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.

翻译：本文提出并分析了用于漂移扩散模型的不连续伽辽金方法。我们探讨了两种不连续伽辽金格式：经典的内罚方法以及节点有界保持方法。虽然内罚方法展示了适定性与收敛性，但无法保证解的非负性。为解决这一缺陷——该性质在实际应用中通常至关重要，我们采用基于凸子集表述的保正性方法，确保解在拉格朗日节点处的非负性。我们通过总结大量数值实验验证了研究结果，突显了该方法在处理载流子输运复杂性方面的新颖性与有效性。