{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,30]],"date-time":"2025-09-30T00:15:43Z","timestamp":1759191343206,"version":"3.37.3"},"reference-count":14,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2019,10,9]],"date-time":"2019-10-09T00:00:00Z","timestamp":1570579200000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2019,10,9]],"date-time":"2019-10-09T00:00:00Z","timestamp":1570579200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100004686","name":"Deanship of Scientific Research, King Faisal University","doi-asserted-by":"publisher","award":["186244"],"award-info":[{"award-number":["186244"]}],"id":[{"id":"10.13039\/501100004686","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2020,9]]},"DOI":"10.1007\/s11075-019-00810-x","type":"journal-article","created":{"date-parts":[[2019,10,9]],"date-time":"2019-10-09T10:45:06Z","timestamp":1570617906000},"page":"209-229","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["An optimal error bound for a finite element approximation of spatially extended predator-prey interaction model"],"prefix":"10.1007","volume":"85","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1111-3054","authenticated-orcid":false,"given":"Ahmed A.","family":"Al Ghafli","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hassan J.","family":"Al Salman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,10,9]]},"reference":[{"key":"810_CR1","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1016\/0022-247X(77)90173-1","volume":"61","author":"RA Adams","year":"1977","unstructured":"Adams, R.A., Fournier, J.: Cone conditions and properties of Sobolev spaces. J. Math. Anal. App1. 61, 713\u2013714 (1977)","journal-title":"J. Math. Anal. App1."},{"key":"810_CR2","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1016\/j.amc.2014.08.050","volume":"246","author":"AA Al Ghafli","year":"2014","unstructured":"Al Ghafli, A.A., Al Salman, H.J.: An improved error bound for a finite element approximation of a reaction-diffusion system of \u03bb \u2212 \u03c9 type. Appl. Math. and Comp. J. 246, 491\u2013501 (2014)","journal-title":"Appl. Math. and Comp. J."},{"key":"810_CR3","first-page":"452","volume-title":"Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems","author":"C Baiocchi","year":"1984","unstructured":"Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, p 452. Wiley, Chichester (1984)"},{"key":"810_CR4","first-page":"1","volume":"72","author":"JW Barrett","year":"1995","unstructured":"Barrett, J.W., Blowey, J.F.: An error bound for the finite element approximation of the Cahn-Hilliard equation with logarithmic free energy. Mumer. Math. 72, 1\u201320 (1995)","journal-title":"Mumer. Math."},{"key":"810_CR5","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1093\/imanum\/18.2.287","volume":"18","author":"JW Barrett","year":"1998","unstructured":"Barrett, J.W., Blowey, J.F.: Finite element approximation of a model for phase separation of a multi-component alloy with a concentration- dependent mobility matrix. IMA J. Mumer. Anal. 18, 287\u2013328 (1998)","journal-title":"IMA J. Mumer. Anal."},{"key":"810_CR6","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1090\/S0025-5718-99-01015-7","volume":"68","author":"JW Barrett","year":"1999","unstructured":"Barrett, J.W., Blowey, J.F.: Finite element approximation of the Cahn-Hilliard equation with logarithmic free energy. Math. Comput. 68, 487\u2013517 (1999)","journal-title":"Math. Comput."},{"issue":"4","key":"810_CR7","first-page":"525","volume":"80","author":"JW Barrett","year":"1999","unstructured":"Barrett, J.W., Blowey, J.F., Gracke, H.: Finite element approximation of a fourth order nonlinear degenerate parabolic equation. M2AN Math. Model. Numer. Ana. 80(4), 525\u2013556 (1999)","journal-title":"M2AN Math. Model. Numer. Ana."},{"key":"810_CR8","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1093\/imanum\/19.1.147","volume":"19","author":"JW Barrett","year":"1999","unstructured":"Barrett, J.W., Blowey, J.F.: An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy. IMA J. Numer Math. 19, 147\u2013168 (1999)","journal-title":"IMA J. Numer Math."},{"issue":"4","key":"810_CR9","first-page":"713","volume":"35","author":"JW Barrett","year":"2001","unstructured":"Barrett, J.W., Blowey, J.F., Gracke, H.: On fully practical finite element approximations of degenerate Cahn-Hilliard systems. Numer. Math. 35(4), 713\u2013748 (2001)","journal-title":"Numer. Math."},{"key":"810_CR10","doi-asserted-by":"publisher","first-page":"621","DOI":"10.1017\/S095679250500639X","volume":"16","author":"MR Garvie","year":"2005","unstructured":"Garvie, M.R., Blowey, J.F.: A reaction-diffusion system of \u03bb \u2212 \u03c9 type. Part II: Numerical analysis. Euro. J. Appl. Math. 16, 621\u2013646 (2005)","journal-title":"Euro. J. Appl. Math."},{"key":"810_CR11","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1007\/s00211-007-0106-x","volume":"107","author":"MR Garvie","year":"2007","unstructured":"Garvie, M.R., Trenchea, C.: Finite element approximation of spatially extended predator-prey interactions with the Holling type II functional response. Numer. Math. 107, 641\u2013667 (2007)","journal-title":"Numer. Math."},{"key":"810_CR12","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1002\/(SICI)1097-0312(200005)53:5<525::AID-CPA1>3.0.CO;2-M","volume":"53","author":"RH Nochetto","year":"2000","unstructured":"Nochetto, R.H., Savar\u00e9, G., Verdi, C.: A posteriori error estimates for variable time-step discretizations of nonlinear evolution equations. Commun. Pure Appl. Math. 53, 525\u2013589 (2000)","journal-title":"Commun. Pure Appl. Math."},{"key":"810_CR13","doi-asserted-by":"publisher","first-page":"948","DOI":"10.1016\/j.camwa.2017.04.030","volume":"74","author":"F Diele","year":"2017","unstructured":"Diele, F., Garvie, M.R., Trenchea, C.: Numerical analysis of a first-order in time implicit-symplectic scheme for predator-prey systems. Comput. Math. Appl. 74, 948\u2013961 (2017)","journal-title":"Comput. Math. Appl."},{"key":"810_CR14","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1016\/j.matcom.2015.02.001","volume":"110","author":"F Diele","year":"2015","unstructured":"Diele, F., Marangi, C., Ragni, S.: IMSP schemes for spatially explicit models of cyclic populations and metapopulation dynamics. Math Comput. Simul. 110, 83\u201395 (2015)","journal-title":"Math Comput. Simul."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00810-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-019-00810-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-019-00810-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,7]],"date-time":"2020-10-07T23:28:01Z","timestamp":1602113281000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-019-00810-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,9]]},"references-count":14,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["810"],"URL":"https:\/\/doi.org\/10.1007\/s11075-019-00810-x","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2019,10,9]]},"assertion":[{"value":"9 October 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 September 2019","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 October 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}