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A Weighted Hybridizable Discontinuous Galerkin Method for Drift-Diffusion Problems

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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 validates 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.

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Funding

This work was partially supported by funds from GNCS-INdAM “Professori Visitatori Bando 2020”, project “A comparison between finite volume and hybridizable discontinuous Galerkin methods for the simulation of micro- and nano-electronic devices”, the Leibniz competition 2020 as well as the National Natural Science Foundation of China (Grant No. 12301496), the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Pisa, CUP I57G22000700001. LH acknowledges partial support from the grant MUR PRIN 2022 No. 2022WKWZA8 “Immersed methods for multiscale and multiphysics problems (IMMEDIATE)”. LH is member of Gruppo Nazionale per il Calcolo Scientifico (GNCS) of Istituto Nazionale di Alta Matematica (INdAM).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, No.2006, Xiyuan Ave, West Hi-Tech Zone, Chengdu, 611731, China

    Wenyu Lei

  2. Mathematics Area, SISSA – International School for Advance Studies, via Bonomea 265, 34136, Trieste, Italy

    Stefano Piani

  3. Weierstrass Institute Berlin, Mohrenstr. 39, 10117, Berlin, Germany

    Patricio Farrell

  4. Dipartimento di Matematica e Informatica, ’Ulisse Dini’ – University of Florence, Viale Morgagni 67/A, 50134, Florence, Italy

    Nella Rotundo

  5. Dipartimento di Matematica, University of Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Luca Heltai

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  1. Wenyu Lei
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Correspondence to Wenyu Lei.

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Lei, W., Piani, S., Farrell, P. et al. A Weighted Hybridizable Discontinuous Galerkin Method for Drift-Diffusion Problems. J Sci Comput 99, 33 (2024). https://doi.org/10.1007/s10915-024-02481-w

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