{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T08:25:15Z","timestamp":1769588715219,"version":"3.49.0"},"reference-count":57,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2022,12,16]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>A diffuse interface model for tumour growth in the presence of a nutrient consumed by the tumour is considered. The system of equations consists of a Cahn\u2013Hilliard equation with source terms for the tumour cells and a reaction\u2013diffusion equation for the nutrient. We introduce a fully-discrete finite element approximation of the model and prove stability bounds for the discrete scheme. Moreover, we show that discrete solutions exist and depend continuously on the initial and boundary data. We then pass to the limit in the discretization parameters and prove convergence to a global-in-time weak solution to the model. Under additional assumptions, this weak solution is unique. Finally, we present some numerical results including numerical error investigation in one spatial dimension and some long time simulations in two and three spatial dimensions.<\/jats:p>","DOI":"10.1515\/jnma-2021-0094","type":"journal-article","created":{"date-parts":[[2022,5,4]],"date-time":"2022-05-04T13:42:06Z","timestamp":1651671726000},"page":"295-324","source":"Crossref","is-referenced-by-count":9,"title":["Numerical analysis for a Cahn\u2013Hilliard system modelling tumour growth with chemotaxis and active transport"],"prefix":"10.1515","volume":"30","author":[{"given":"Harald","family":"Garcke","sequence":"first","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik , Universit\u00e4t Regensburg , 93053 , Regensburg , Germany"}]},{"given":"Dennis","family":"Trautwein","sequence":"additional","affiliation":[{"name":"Fakult\u00e4t f\u00fcr Mathematik , Universit\u00e4t Regensburg , 93053 , Regensburg , Germany"}]}],"member":"374","published-online":{"date-parts":[[2022,5,5]]},"reference":[{"key":"2024032505541370045_j_jnma-2021-0094_ref_001","doi-asserted-by":"crossref","unstructured":"H. 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