A weighted Hybridizable Discontinuous Galerkin method for drift-diffusion problems

TL;DR

This paper introduces a weighted hybridizable discontinuous Galerkin (W-HDG) method for drift-diffusion problems that generalizes the Scharfetter-Gummel scheme to high-order accuracy using exponential weights, ensuring optimal convergence.

Contribution

The paper develops a novel W-HDG scheme that mimics Slotboom variables, eliminates the drift term locally, and extends the Scharfetter-Gummel method to higher polynomial degrees.

Findings

The scheme is well-posed and numerically stable.

It achieves optimal convergence and superconvergence of postprocessed solutions.

For specific parameters, it reduces to the classical Scharfetter-Gummel scheme.

Abstract

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.