{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:47:02Z","timestamp":1760402822750,"version":"build-2065373602"},"reference-count":58,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,3,30]],"date-time":"2021-03-30T00:00:00Z","timestamp":1617062400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009117","name":"Technische Universit\u00e4t Chemnitz","doi-asserted-by":"publisher","award":["ID 12-2021"],"award-info":[{"award-number":["ID 12-2021"]}],"id":[{"id":"10.13039\/100009117","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Three-component systems of diffusion\u2013reaction equations play a central role in the modelling and simulation of chemical processes in engineering, electro-chemistry, physical chemistry, biology, population dynamics, etc. A major question in the simulation of three-component systems is how to guarantee non-negative species distributions in the model and how to calculate them effectively. Current numerical methods to enforce non-negative species distributions tend to be cost-intensive in terms of computation time and they are not robust for big rate constants of the considered reaction. In this article, a method, as a combination of homotopy methods, modern augmented Lagrangian methods, and adaptive FEMs is outlined to obtain a robust and efficient method to simulate diffusion\u2013reaction models with non-negative concentrations. Although in this paper the convergence analysis is not described rigorously, multiple numerical examples as well as an application to elctro-deposition from an aqueous Cu2+-(\u03b2-alanine) electrolyte are presented.<\/jats:p>","DOI":"10.3390\/a14040113","type":"journal-article","created":{"date-parts":[[2021,3,30]],"date-time":"2021-03-30T09:57:04Z","timestamp":1617098224000},"page":"113","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On a Robust and Efficient Numerical Scheme for the Simulation of Stationary 3-Component Systems with Non-Negative Species-Concentration with an Application to the Cu Deposition from a Cu-(\u03b2-alanine)-Electrolyte"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9229-8793","authenticated-orcid":false,"given":"Stephan Daniel","family":"Schwoebel","sequence":"first","affiliation":[{"name":"Materials and Surface Engineering Group, Institute of Materials Science and Engineering, Chemnitz University of Technology, D-09107 Chemnitz, Germany"}]},{"given":"Thomas","family":"Mehner","sequence":"additional","affiliation":[{"name":"Materials and Surface Engineering Group, Institute of Materials Science and Engineering, Chemnitz University of Technology, D-09107 Chemnitz, Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2390-9159","authenticated-orcid":false,"given":"Thomas","family":"Lampke","sequence":"additional","affiliation":[{"name":"Materials and Surface Engineering Group, Institute of Materials Science and Engineering, Chemnitz University of Technology, D-09107 Chemnitz, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,30]]},"reference":[{"key":"ref_1","unstructured":"Newman, J.S. 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