In this work we propose a weighted hybridizable discontinuous Galerkin method (W-HDG) for drift-diffusion problems. By using specific exponential weights when computing the $L^2$ product in each cell of the discretization, we are able to mimic the behavior of the Slotboom variables, and eliminate the drift term from the local matrix contributions, while still solving the problem for the primal variables. We show that the proposed numerical scheme is well-posed, and validate numerically that it has the same properties as classical HDG methods, including optimal convergence, and superconvergence of postprocessed solutions. For polynomial degree zero, dimension one, and vanishing HDG stabilization parameter, W-HDG coincides with the Scharfetter-Gummel finite volume scheme (i.e., it produces the same system matrix). The use of local exponential weights generalizes the Scharfetter-Gummel scheme (the state-of-the-art for finite volume discretization of transport dominated problems) to arbitrary high order approximations.

翻译：本文提出了一种用于漂移-扩散问题的加权混合间断伽辽金方法（W-HDG）。通过在离散单元的$L^2$内积计算中引入特定的指数权重，我们能够模拟Slotboom变量的行为，从而在保持求解原始变量的同时消除局部矩阵中的漂移项贡献。我们证明了所提出的数值格式是适定的，并通过数值验证表明其具备经典HDG方法的相同性质，包括最优收敛性和后处理解的超收敛性。当多项式次数为零、维度为一且HDG稳定化参数趋于零时，W-HDG退化为Scharfetter-Gummel有限体积格式（即生成相同的系统矩阵）。局部指数权重的引入将Scharfetter-Gummel格式（当前传输主导问题有限体积离散的先进方案）推广至任意高阶近似。