{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:39:26Z","timestamp":1740123566920,"version":"3.37.3"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,3,21]],"date-time":"2024-03-21T00:00:00Z","timestamp":1710979200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,3,21]],"date-time":"2024-03-21T00:00:00Z","timestamp":1710979200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2024,5]]},"DOI":"10.1007\/s10915-024-02481-w","type":"journal-article","created":{"date-parts":[[2024,3,21]],"date-time":"2024-03-21T10:01:58Z","timestamp":1711015318000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Weighted Hybridizable Discontinuous Galerkin Method for Drift-Diffusion Problems"],"prefix":"10.1007","volume":"99","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3239-7124","authenticated-orcid":false,"given":"Wenyu","family":"Lei","sequence":"first","affiliation":[]},{"given":"Stefano","family":"Piani","sequence":"additional","affiliation":[]},{"given":"Patricio","family":"Farrell","sequence":"additional","affiliation":[]},{"given":"Nella","family":"Rotundo","sequence":"additional","affiliation":[]},{"given":"Luca","family":"Heltai","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,3,21]]},"reference":[{"key":"2481_CR1","doi-asserted-by":"publisher","DOI":"10.1016\/j.electacta.2021.138696","volume":"390","author":"D Abdel","year":"2021","unstructured":"Abdel, D., Vagner, P., Fuhrmann, J., Farrell, P.: Modelling charge transport in perovskite solar cells: potential-based and limiting ion vacancy depletion. Electrochim. Acta 390, 138696 (2021)","journal-title":"Electrochim. Acta"},{"issue":"12","key":"2481_CR2","doi-asserted-by":"publisher","first-page":"4666","DOI":"10.1016\/j.na.2010.02.046","volume":"72","author":"G Al\u00ec","year":"2010","unstructured":"Al\u00ec, G., Rotundo, N.: An existence result for elliptic partial differential\u2013algebraic equations arising in semiconductor modeling. Nonlinear Anal. Theory Methods Appl. 72(12), 4666\u20134681 (2010)","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"issue":"09","key":"2481_CR3","doi-asserted-by":"publisher","first-page":"1261","DOI":"10.1142\/S0218202503002908","volume":"13","author":"G Al\u00ec","year":"2003","unstructured":"Al\u00ec, G., Bartel, A., G\u00fcnther, M., Tischendorf, C.: Elliptic partial differential-algebraic multiphysics models in electrical network design. Math. Models Methods Appl. Sci. 13(09), 1261\u20131278 (2003)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"3","key":"2481_CR4","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1515\/jnma-2021-0081","volume":"29","author":"D Arndt","year":"2021","unstructured":"Arndt, D., Bangerth, W., Blais, B., Fehling, M., Gassm\u00f6ller, R., Heister, T., Heltai, L., K\u00f6cher, U., Kronbichler, M., Maier, M., Munch, P., Pelteret, J.P., Proell, S., Simon, K., Turcksin, B., Wells, D., Zhang, J.: The deal.II library, version 9.3. J. Numer. Math. 29(3), 171\u2013186 (2021)","journal-title":"J. Numer. Math."},{"key":"2481_CR5","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1016\/j.camwa.2020.02.022","volume":"81","author":"D Arndt","year":"2021","unstructured":"Arndt, D., Bangerth, W., Davydov, D., Heister, T., Heltai, L., Kronbichler, M., Maier, M., Pelteret, J.P., Turcksin, B., Wells, D.: The deal. II finite element library: design, features, and insights. Comput. Math. Appl. 81, 407\u2013422 (2021)","journal-title":"Comput. Math. Appl."},{"issue":"5","key":"2481_CR6","doi-asserted-by":"publisher","first-page":"1749","DOI":"10.1137\/S0036142901384162","volume":"39","author":"DN Arnold","year":"2001","unstructured":"Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749\u20131779 (2001)","journal-title":"SIAM J. Numer. Anal."},{"key":"2481_CR7","doi-asserted-by":"publisher","first-page":"777","DOI":"10.1137\/0724050","volume":"24","author":"RE Bank","year":"1987","unstructured":"Bank, R.E., Rose, D.J.: Some error estimates for the box method. SIAM J. Numer. Anal. 24, 777\u2013787 (1987)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"2481_CR8","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1007\/s00211-012-0448-x","volume":"121","author":"M Bessemoulin-Chatard","year":"2012","unstructured":"Bessemoulin-Chatard, M.: A finite volume scheme for convection-diffusion equations with nonlinear diffusion derived from the Scharfetter-Gummel scheme. Numer. Math. 121(4), 637\u2013670 (2012)","journal-title":"Numer. Math."},{"key":"2481_CR9","volume-title":"Mixed and Hybrid Finite Element Methods","author":"F Brezzi","year":"2012","unstructured":"Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, vol. 15. Springer, Berlin (2012)"},{"issue":"1\u20133","key":"2481_CR10","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1016\/0045-7825(89)90044-3","volume":"75","author":"F Brezzi","year":"1989","unstructured":"Brezzi, F., Marini, L.D., Pietra, P.: Numerical simulation of semiconductor devices. Comput. Methods Appl. Mech. Eng. 75(1\u20133), 493\u2013514 (1989)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"6","key":"2481_CR11","doi-asserted-by":"publisher","first-page":"1342","DOI":"10.1137\/0726078","volume":"26","author":"F Brezzi","year":"1989","unstructured":"Brezzi, F., Marini, L.D., Pietra, P.: Two-dimensional exponential fitting and applications to drift-diffusion models. SIAM J. Numer. Anal. 26(6), 1342\u20131355 (1989)","journal-title":"SIAM J. Numer. Anal."},{"key":"2481_CR12","doi-asserted-by":"crossref","unstructured":"Chainais-Hillairet, C., J\u00fcngel, A., Shpartko, P.: A finite-volume scheme for a spinorial matrix drift-diffusion model for semiconductors. Numerical Methods for Partial Differential Equations (2015)","DOI":"10.1002\/num.22030"},{"issue":"4","key":"2481_CR13","doi-asserted-by":"publisher","first-page":"963","DOI":"10.1007\/s00211-022-01289-w","volume":"151","author":"C Chainais-Hillairet","year":"2022","unstructured":"Chainais-Hillairet, C., Herda, M., Lemaire, S., Moatti, J.: Long-time behaviour of hybrid finite volume schemes for advection-diffusion equations: linear and nonlinear approaches. Numer. Math. 151(4), 963\u20131016 (2022). https:\/\/doi.org\/10.1007\/s00211-022-01289-w","journal-title":"Numer. Math."},{"issue":"1","key":"2481_CR14","doi-asserted-by":"publisher","first-page":"420","DOI":"10.1007\/s10915-019-00945-y","volume":"80","author":"G Chen","year":"2019","unstructured":"Chen, G., Monk, P., Zhang, Y.: An HDG method for the time-dependent drift-diffusion model of semiconductor devices. J. Sci. Comput. 80(1), 420\u2013443 (2019)","journal-title":"J. Sci. Comput."},{"key":"2481_CR15","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719208","volume-title":"The finite element method for elliptic problems","author":"PG Ciarlet","year":"2002","unstructured":"Ciarlet, P.G.: The finite element method for elliptic problems. SIAM, Philadelphia (2002)"},{"issue":"264","key":"2481_CR16","doi-asserted-by":"publisher","first-page":"1887","DOI":"10.1090\/S0025-5718-08-02123-6","volume":"77","author":"B Cockburn","year":"2008","unstructured":"Cockburn, B., Dong, B., Guzm\u00e1n, J.: A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77(264), 1887\u20131916 (2008)","journal-title":"Math. Comput."},{"issue":"5","key":"2481_CR17","doi-asserted-by":"publisher","first-page":"3827","DOI":"10.1137\/080728810","volume":"31","author":"B Cockburn","year":"2009","unstructured":"Cockburn, B., Dong, B., Guzm\u00e1n, J., Restelli, M., Sacco, R.: A hybridizable discontinuous Galerkin method for steady-state convection-diffusion-reaction problems. SIAM J. Sci. Comput. 31(5), 3827\u20133846 (2009)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"2481_CR18","doi-asserted-by":"publisher","first-page":"1319","DOI":"10.1137\/070706616","volume":"47","author":"B Cockburn","year":"2009","unstructured":"Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47(2), 1319\u20131365 (2009)","journal-title":"SIAM J. Numer. Anal."},{"issue":"265","key":"2481_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0025-5718-08-02146-7","volume":"78","author":"B Cockburn","year":"2009","unstructured":"Cockburn, B., Guzm\u00e1n, J., Wang, H.: Superconvergent discontinuous Galerkin methods for second-order elliptic problems. Math. Comput. 78(265), 1\u201324 (2009)","journal-title":"Math. Comput."},{"issue":"5","key":"2481_CR20","doi-asserted-by":"publisher","first-page":"2135","DOI":"10.1137\/140993971","volume":"53","author":"DA Di Pietro","year":"2015","unstructured":"Di Pietro, D.A., Droniou, J., Ern, A.: A discontinuous-skeletal method for advection-diffusion-reaction on general meshes. SIAM J. Numer. Anal. 53(5), 2135\u20132157 (2015). https:\/\/doi.org\/10.1137\/140993971","journal-title":"SIAM J. Numer. Anal."},{"issue":"02","key":"2481_CR21","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1142\/S0218202510004222","volume":"20","author":"J Droniou","year":"2010","unstructured":"Droniou, J., Eymard, R., Gallou\u00ebt, T., Herbin, R.: A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. 20(02), 265\u2013295 (2010)","journal-title":"Math. Models Methods Appl. Sci."},{"key":"2481_CR22","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-4355-5","volume-title":"Theory and Practice of Finite Elements","author":"A Ern","year":"2004","unstructured":"Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements, vol. 159. Springer, Berlin (2004)"},{"issue":"4","key":"2481_CR23","doi-asserted-by":"publisher","first-page":"1009","DOI":"10.1093\/imanum\/drn084","volume":"30","author":"R Eymard","year":"2009","unstructured":"Eymard, R., Gallou\u00ebt, T., Herbin, R.: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal. 30(4), 1009\u20131043 (2009). https:\/\/doi.org\/10.1093\/imanum\/drn084","journal-title":"IMA J. Numer. Anal."},{"key":"2481_CR24","doi-asserted-by":"crossref","unstructured":"Farrell, P., Rotundo, N., Doan, D.H., Kantner, M., Fuhrmann, J., Koprucki, T.: Mathematical Methods: Drift-Diffusion Models. In: J.\u00a0Piprek (ed.) Handbook of Optoelectronic Device Modeling and Simulation, vol.\u00a02, chap.\u00a050, pp. 733\u2013771. CRC Press, Boca Raton (2017)","DOI":"10.4324\/9781315152318-25"},{"issue":"1","key":"2481_CR25","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1007\/s10915-017-0361-7","volume":"72","author":"P Farrell","year":"2017","unstructured":"Farrell, P., Linke, A.: Uniform second order convergence of a complete flux scheme on unstructured 1d grids for a singularly perturbed advection-diffusion equation and some multidimensional extensions. J. Sci. Comput. 72(1), 373\u2013395 (2017)","journal-title":"J. Sci. Comput."},{"key":"2481_CR26","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1016\/j.jcp.2017.06.023","volume":"346","author":"P Farrell","year":"2017","unstructured":"Farrell, P., Koprucki, T., Fuhrmann, J.: Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond boltzmann statistics. J. Comput. Phys. 346, 497\u2013513 (2017)","journal-title":"J. Comput. Phys."},{"issue":"3","key":"2481_CR27","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1007\/s10825-015-0712-2","volume":"14","author":"K G\u00e4rtner","year":"2015","unstructured":"G\u00e4rtner, K.: Existence of bounded discrete steady state solutions of the van Roosbroeck system with monotone Fermi-Dirac statistic functions. J. Comput. Electron. 14(3), 773\u2013787 (2015)","journal-title":"J. Comput. Electron."},{"key":"2481_CR28","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611972030","volume-title":"Elliptic problems in nonsmooth domains","author":"P Grisvard","year":"2011","unstructured":"Grisvard, P.: Elliptic problems in nonsmooth domains. SIAM, Philadelphia (2011)"},{"key":"2481_CR29","unstructured":"Lazarov, R.D., Zikatanov, L.T.: An exponential fitting scheme for general convection-diffusion equations on tetrahedral meshes. arXiv preprint arXiv:1211.0869 (2012)"},{"key":"2481_CR30","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1016\/j.cam.2013.03.011","volume":"250","author":"L Liu","year":"2013","unstructured":"Liu, L., van Dijk, J., ten Thije Boonkkamp, J., Mihailova, D., van der Mullen, J.: The complete flux scheme\u2013error analysis and application to plasma simulation. J. Comput. Appl. Math. 250, 229\u2013243 (2013)","journal-title":"J. Comput. Appl. Math."},{"key":"2481_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7091-3678-2","volume-title":"The Stationary Semiconductor Device Equations","author":"PA Markowich","year":"1986","unstructured":"Markowich, P.A.: The Stationary Semiconductor Device Equations. Springer, Vienna (1986)"},{"key":"2481_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-7091-3678-2","volume-title":"The Stationary Semiconductor Device Equations","author":"PA Markowich","year":"1986","unstructured":"Markowich, P.A.: The Stationary Semiconductor Device Equations. Springer, Berlin (1986)"},{"issue":"6","key":"2481_CR33","doi-asserted-by":"publisher","first-page":"781","DOI":"10.1002\/cpa.3160250606","volume":"25","author":"MS Mock","year":"1972","unstructured":"Mock, M.S.: On equations describing steady-state carrier distributions in a semiconductor device. Commun. Pure Appl. Math. 25(6), 781\u2013792 (1972)","journal-title":"Commun. Pure Appl. Math."},{"key":"2481_CR34","volume-title":"Numerical Solution of Convection-Diffusion Problems","author":"K Morton","year":"1996","unstructured":"Morton, K.: Numerical Solution of Convection-Diffusion Problems. Applied Mathematics. Taylor & Francis, Milton Park (1996)"},{"issue":"9","key":"2481_CR35","doi-asserted-by":"publisher","first-page":"3232","DOI":"10.1016\/j.jcp.2009.01.030","volume":"228","author":"NC Nguyen","year":"2009","unstructured":"Nguyen, N.C., Peraire, J., Cockburn, B.: An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations. J. Comput. Phys. 228(9), 3232\u20133254 (2009)","journal-title":"J. Comput. Phys."},{"key":"2481_CR36","unstructured":"Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer Series in Computational Mathematics vol. 24, 2nd edn. Springer, Berlin (2008)"},{"issue":"1","key":"2481_CR37","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1109\/T-ED.1969.16566","volume":"16","author":"D Scharfetter","year":"1969","unstructured":"Scharfetter, D., Gummel, H.: Large-signal analysis of a silicon read diode oscillator. IEEE Trans. Electron Devices 16(1), 64\u201377 (1969)","journal-title":"IEEE Trans. Electron Devices"},{"key":"2481_CR38","unstructured":"Shewchuk, J.: Triangle: a two-dimensional quality mesh generator and Delaunay triangulator. http:\/\/www.cs.cmu.edu\/~quake\/triangle.html, University of California at Berkeley"},{"issue":"2","key":"2481_CR39","doi-asserted-by":"publisher","first-page":"11:1","DOI":"10.1145\/2629697","volume":"41","author":"H Si","year":"2015","unstructured":"Si, H.: Tetgen, a delaunay-based quality tetrahedral mesh generator. ACM Trans. Math. Softw. 41(2), 11:1-11:36 (2015)","journal-title":"ACM Trans. Math. Softw."},{"issue":"8","key":"2481_CR40","doi-asserted-by":"publisher","first-page":"669","DOI":"10.1109\/t-ed.1973.17727","volume":"20","author":"J Slotboom","year":"1973","unstructured":"Slotboom, J.: Computer-aided two-dimensional analysis of bipolar transistors. IEEE Trans. Electron. Devices 20(8), 669\u2013679 (1973). https:\/\/doi.org\/10.1109\/t-ed.1973.17727","journal-title":"IEEE Trans. Electron. Devices"},{"key":"2481_CR41","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-7049-7","volume-title":"Partial Differential Equations III","author":"ME Taylor","year":"2011","unstructured":"Taylor, M.E.: Partial Differential Equations III. Springer, New York (2011)"},{"issue":"1","key":"2481_CR42","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1007\/s10915-010-9388-8","volume":"46","author":"J ten Thije Boonkkamp","year":"2011","unstructured":"ten Thije Boonkkamp, J., Anthonissen, M.: The finite volume-complete flux scheme for advection-diffusion-reaction equations. J. Sci. Comput. 46(1), 47\u201370 (2011)","journal-title":"J. Sci. Comput."},{"issue":"2","key":"2481_CR43","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1108\/eb010116","volume":"12","author":"J Ten Thije Boonkkamp","year":"1993","unstructured":"Ten Thije Boonkkamp, J., Schilders, W.H.: An exponential fitting scheme for the electrothermal device equations specifically for the simulation of avalanche generation. COMPEL-The Int. J. Comput. Math. Electric. Electron. Eng. 12(2), 95\u2013111 (1993)","journal-title":"COMPEL-The Int. J. Comput. Math. Electric. Electron. Eng."},{"key":"2481_CR44","unstructured":"ten Thije\u00a0Boonkkamp, J.: A complete flux scheme for one-dimensional combustion simulation. In: Finite volumes for complex applications IV, pp. 573\u2013583. ISTE, London (2005)"},{"issue":"1","key":"2481_CR45","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1080\/10407799008944943","volume":"18","author":"GD Thiart","year":"1990","unstructured":"Thiart, G.D.: Improved finite-difference scheme for the solution of convection-diffusion problems with the simplen algorithm. Numer. Heat Transf. Part B 18(1), 81\u201395 (1990)","journal-title":"Numer. Heat Transf. Part B"},{"key":"2481_CR46","doi-asserted-by":"crossref","unstructured":"van\u2019t Hof, B., ten Thije\u00a0Boonkkamp, J.H.M., Mattheij, R.M.M.: Discretization of the stationary convection-diffusion-reaction equation. Numer. Methods Partial Differential Equations 14(5), 607\u2013625 (1998)","DOI":"10.1002\/(SICI)1098-2426(199809)14:5<607::AID-NUM5>3.0.CO;2-M"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-024-02481-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-024-02481-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-024-02481-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,30]],"date-time":"2024-04-30T19:05:08Z","timestamp":1714503908000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-024-02481-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,21]]},"references-count":46,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,5]]}},"alternative-id":["2481"],"URL":"https:\/\/doi.org\/10.1007\/s10915-024-02481-w","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"type":"print","value":"0885-7474"},{"type":"electronic","value":"1573-7691"}],"subject":[],"published":{"date-parts":[[2024,3,21]]},"assertion":[{"value":"26 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 January 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 February 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 March 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"33"}}