{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T21:42:57Z","timestamp":1780350177247,"version":"3.54.1"},"reference-count":42,"publisher":"American Mathematical Society (AMS)","issue":"356","license":[{"start":{"date-parts":[[2025,11,27]],"date-time":"2025-11-27T00:00:00Z","timestamp":1764201600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100005230","name":"Natural Science Foundation of Chongqing Municipality","doi-asserted-by":"publisher","award":["CSTB2024NSCQ-MSX0221"],"award-info":[{"award-number":["CSTB2024NSCQ-MSX0221"]}],"id":[{"id":"10.13039\/501100005230","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper, we prove the optimal error estimate of an unconditionally positivity-preserving, mass-conserving and energy stable method for the Keller-Segel chemotaxis equations. Applying a log-transformation to preserve the positivity and utilizing a recovery to ensure the mass-conversing, we consider a decoupled and linear fully discrete finite element method for the Keller-Segel chemotaxis equations. Then, supposing that the initial mass is less than certain critical threshold which guarantees that the system does not blow up at a finite time, and deriving the errors of the temporal and spatial discretizations in a proper sequence to avoid extra regularity requirements of the weak solutions, we prove that the method is unconditionally energy stable and can achieve the optimal convergence order in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    norm under weaker assumptions on the solutions than the existent ones. The shown numerical examples confirm the correctness of the theoretical prediction.\n                  <\/p>","DOI":"10.1090\/mcom\/4041","type":"journal-article","created":{"date-parts":[[2024,10,23]],"date-time":"2024-10-23T13:31:10Z","timestamp":1729690270000},"page":"2761-2793","source":"Crossref","is-referenced-by-count":3,"title":["Optimal error estimate of unconditionally positivity-preserving, mass-conserving and energy stable method for the Keller-Segel chemotaxis model"],"prefix":"10.1090","volume":"94","author":[{"given":"Kun","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Enlong","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinlong","family":"Feng","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2024,11,27]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"issue":"9","key":"2","doi-asserted-by":"publisher","first-page":"1663","DOI":"10.1142\/S021820251550044X","article-title":"Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues","volume":"25","author":"Bellomo, N.","year":"2015","journal-title":"Math. 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