{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T06:24:21Z","timestamp":1770963861117,"version":"3.50.1"},"reference-count":42,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T00:00:00Z","timestamp":1770422400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"publisher","award":["2025D01B197"],"award-info":[{"award-number":["2025D01B197"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Research Foundation of China University of Petroleum\u2013Beijing at Karamay","award":["XQZX20230030"],"award-info":[{"award-number":["XQZX20230030"]}]},{"name":"Xinjiang University Doctoral Science and Technology Innovation Program","award":["XJU2023BS026"],"award-info":[{"award-number":["XJU2023BS026"]}]},{"name":"Tianyuan Fund for Mathematics of the National Natural Science Foundation of China","award":["12326102"],"award-info":[{"award-number":["12326102"]}]},{"name":"2025 National Undergraduate Innovation and Entrepreneurship Training Program of China","award":["202519414009"],"award-info":[{"award-number":["202519414009"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Designing structure-preserving numerical schemes for the generalized Poisson-Nernst-Planck (PNP) system is challenging due to its inherent strong nonlinearity and coupling. In this paper, we propose a class of efficient, unconditional energy-stable schemes based on the Stabilized Scalar Auxiliary Variable (S-SAV) framework combined with the finite element method. We construct both first-order (BE-S-SAV) and second-order (BDF2-S-SAV) fully discrete schemes. A distinguishing feature of our approach is the use of a linear decomposition strategy, which decouples the complex nonlinear system into a sequence of linear, constant-coefficient elliptic equations at each time step. This significantly reduces computational complexity by avoiding expensive nonlinear iterations. We provide rigorous theoretical proofs demonstrating that the proposed schemes are unconditionally energy stable and strictly preserve mass conservation. Numerical experiments satisfy the theoretical analysis, confirming optimal convergence rates and demonstrating robust preservation of mass conservation and modified energy stability in the tested regimes.<\/jats:p>","DOI":"10.3390\/axioms15020126","type":"journal-article","created":{"date-parts":[[2026,2,9]],"date-time":"2026-02-09T08:15:54Z","timestamp":1770624954000},"page":"126","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Energy-Stable S-SAV Finite Element Method for the Generalized Poisson-Nernst-Planck Equation"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-5312-7405","authenticated-orcid":false,"given":"Maoqin","family":"Yuan","sequence":"first","affiliation":[{"name":"School of Science and Arts, China University of Petroleum-Beijing at Karamay, Karamay 834000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-9652-6984","authenticated-orcid":false,"given":"Junde","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Science and Arts, China University of Petroleum-Beijing at Karamay, Karamay 834000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peng","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Science and Arts, China University of Petroleum-Beijing at Karamay, Karamay 834000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-4315-9772","authenticated-orcid":false,"given":"Mingyang","family":"Li","sequence":"additional","affiliation":[{"name":"Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China"},{"name":"University of Chinese Academy of Sciences, Beijing 100049, China"},{"name":"State Key Laboratory of Advanced Manufacturing for Optical Systems, Changchun 130033, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0022-0728(78)80137-5","article-title":"Numerical solution of the Nernst\u2013Planck and Poisson equation system with applications to membrane electrochemistry and solid state physics","volume":"90","author":"Brumleve","year":"1978","journal-title":"J. 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