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While recent theoretical advancements have established the critical dynamics of the harmonic component in such flows, practical applications remain computationally restricted by a lack of spatial and temporal adaptivity. Furthermore, simulations on poor\u2010quality meshes often lead to numerical instability and a failure to preserve the flow's underlying harmonic component when using naive interpolation methods. In this paper, we introduce Adaptive Fluid Cohomology, a framework that integrates dynamic spatial and temporal refinement into the simulation of the Euler equations. We leverage a posteriori error estimation to adjust spatial resolution on the fly, alongside a standard Dormand\u2010Prince 5(4) time\u2010stepping scheme for temporal accuracy. To ensure stability during mesh mutations, we develop a novel method that robustly transfers the harmonic basis during remeshing. While our experimental evaluation focuses on 2D surface flows, the underlying theoretical formulation is presented to capture the 3D setting as well. Our evaluation demonstrates that this adaptive approach accurately recreates the dynamics of high\u2010resolution simulations while reducing the memory footprint by up to\n                    <jats:italic>86%<\/jats:italic>\n                    and maintaining numerical stability even on poor\u2010quality triangulations where static methods fail.\n                  <\/jats:p>","DOI":"10.1111\/cgf.70569","type":"journal-article","created":{"date-parts":[[2026,8,12]],"date-time":"2026-08-12T10:58:45Z","timestamp":1786532325000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Adaptive Fluid Cohomology on Surfaces"],"prefix":"10.1111","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-4076-5209","authenticated-orcid":false,"given":"Bastian","family":"Abt","sequence":"first","affiliation":[{"name":"University of Bonn"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-2270-5710","authenticated-orcid":false,"given":"David","family":"Stotko","sequence":"additional","affiliation":[{"name":"University of Bonn"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7787-3622","authenticated-orcid":false,"given":"Nils","family":"Wandel","sequence":"additional","affiliation":[{"name":"University of Bonn"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5505-9347","authenticated-orcid":false,"given":"Reinhard","family":"Klein","sequence":"additional","affiliation":[{"name":"University of Bonn"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,8,12]]},"reference":[{"key":"e_1_2_8_2_2","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-fluid-121108-145519"},{"key":"e_1_2_8_3_2","doi-asserted-by":"publisher","DOI":"10.1145\/2019406.2019408"},{"key":"e_1_2_8_4_2","doi-asserted-by":"publisher","DOI":"10.1145\/2461912.2461982"},{"key":"e_1_2_8_5_2","doi-asserted-by":"publisher","DOI":"10.1145\/1276377.1276502"},{"key":"e_1_2_8_6_2","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.2000.1132"},{"key":"e_1_2_8_7_2","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1017\/S0962492924000011","article-title":"Adaptive finite element methods","volume":"33","author":"Bonito A.","year":"2024","journal-title":"Acta Numerica"},{"key":"e_1_2_8_8_2","doi-asserted-by":"publisher","DOI":"10.1175\/1520\u20100485(1999)029<1239:AMRFFD>2.0.CO;2"},{"key":"e_1_2_8_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/0010-4655(88)90020-3"},{"key":"e_1_2_8_10_2","doi-asserted-by":"publisher","DOI":"10.1109\/TVCG.2012.316"},{"key":"e_1_2_8_11_2","volume-title":"Computational electromagnetism: variational formulations, complementarity, edge elements","author":"Bossavit A.","year":"1998"},{"key":"e_1_2_8_12_2","unstructured":"BarnatA. 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