{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T00:56:49Z","timestamp":1772499409553,"version":"3.50.1"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2019,12,14]],"date-time":"2019-12-14T00:00:00Z","timestamp":1576281600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,12,14]],"date-time":"2019-12-14T00:00:00Z","timestamp":1576281600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000765","name":"University College London","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100000765","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2020,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The numerical approximation of an inverse problem subject to the convection\u2013diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are of a form suitable for use in numerical analysis and with explicit dependence on the P\u00e9clet number. A stabilized finite element method is then proposed and analysed. An upper bound on the condition number is first derived. Combining the stability estimates on the continuous problem with the numerical stability of the method, we then obtain error estimates in local <jats:inline-formula><jats:alternatives><jats:tex-math>$$H^1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>H<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>- or <jats:inline-formula><jats:alternatives><jats:tex-math>$$L^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>L<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-norms that are optimal with respect to the approximation order, the problem\u2019s stability and perturbations in data. The convergence order is the same for both norms, but the <jats:inline-formula><jats:alternatives><jats:tex-math>$$H^1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>H<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-estimate requires an additional divergence assumption for the convective field. The theory is illustrated in some computational examples.\n<\/jats:p>","DOI":"10.1007\/s00211-019-01087-x","type":"journal-article","created":{"date-parts":[[2019,12,16]],"date-time":"2019-12-16T15:56:39Z","timestamp":1576511799000},"page":"451-477","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["A stabilized finite element method for inverse problems subject to the convection\u2013diffusion equation. I: diffusion-dominated regime"],"prefix":"10.1007","volume":"144","author":[{"given":"Erik","family":"Burman","sequence":"first","affiliation":[]},{"given":"Mihai","family":"Nechita","sequence":"additional","affiliation":[]},{"given":"Lauri","family":"Oksanen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,12,14]]},"reference":[{"key":"1087_CR1","doi-asserted-by":"crossref","first-page":"123004","DOI":"10.1088\/0266-5611\/25\/12\/123004","volume":"25","author":"G Alessandrini","year":"2009","unstructured":"Alessandrini, G., Rondi, L., Rosset, E., Vessella, S.: The stability for the Cauchy problem for elliptic equations. Inverse Probl. 25, 123004 (2009)","journal-title":"Inverse Probl."},{"key":"1087_CR2","doi-asserted-by":"publisher","first-page":"035004","DOI":"10.1088\/1361-6420\/aaa32b","volume":"34","author":"E Burman","year":"2018","unstructured":"Burman, E., Hansbo, P., Larson, M.G.: Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization. Inverse Probl. 34, 035004 (2018)","journal-title":"Inverse Probl."},{"key":"1087_CR3","unstructured":"Burman, E., Nechita, M., Oksanen, L.: A stabilized finite element method for inverse problems subject to the convection\u2013diffusion equation. II: convection-dominated regime. in preparation, 2019"},{"key":"1087_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.matpur.2018.10.003","volume":"129","author":"E Burman","year":"2019","unstructured":"Burman, E., Nechita, M., Oksanen, L.: Unique continuation for the Helmholtz equation using stabilized finite element methods. J. Math. Pures Appl. 129, 1\u201324 (2019)","journal-title":"J. Math. Pures Appl."},{"issue":"3","key":"1087_CR5","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1007\/s00211-018-0949-3","volume":"139","author":"E Burman","year":"2018","unstructured":"Burman, E., Oksanen, L.: Data assimilation for the heat equation using stabilized finite element methods. Numer. Math. 139(3), 505\u2013528 (2018)","journal-title":"Numer. Math."},{"issue":"1","key":"1087_CR6","doi-asserted-by":"publisher","first-page":"306","DOI":"10.1137\/S0036142902401311","volume":"41","author":"SC Brenner","year":"2003","unstructured":"Brenner, S.C.: Poincar\u00e9\u2013Friedrichs inequalities for piecewise $$H^1$$ functions. SIAM J. Numer. Anal. 41(1), 306\u2013324 (2003)","journal-title":"SIAM J. Numer. Anal."},{"issue":"5","key":"1087_CR7","doi-asserted-by":"publisher","first-page":"2012","DOI":"10.1137\/S0036142903437374","volume":"43","author":"E Burman","year":"2005","unstructured":"Burman, E.: A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal. 43(5), 2012\u20132033 (2005)","journal-title":"SIAM J. Numer. Anal."},{"issue":"6","key":"1087_CR8","doi-asserted-by":"publisher","first-page":"A2752","DOI":"10.1137\/130916862","volume":"35","author":"E Burman","year":"2013","unstructured":"Burman, E.: Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: elliptic equations. SIAM J. Sci. Comput. 35(6), A2752\u2013A2780 (2013)","journal-title":"SIAM J. Sci. Comput."},{"issue":"7\u20138","key":"1087_CR9","doi-asserted-by":"publisher","first-page":"655","DOI":"10.1016\/j.crma.2014.06.008","volume":"352","author":"E Burman","year":"2014","unstructured":"Burman, E.: Error estimates for stabilized finite element methods applied to ill-posed problems. C. R. Math. Acad. Sci. Paris 352(7\u20138), 655\u2013659 (2014)","journal-title":"C. R. Math. Acad. Sci. Paris"},{"issue":"3","key":"1087_CR10","doi-asserted-by":"publisher","first-page":"349","DOI":"10.1007\/s00211-007-0067-0","volume":"106","author":"R Becker","year":"2007","unstructured":"Becker, R., Vexler, B.: Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106(3), 349\u2013367 (2007)","journal-title":"Numer. Math."},{"issue":"5","key":"1087_CR11","doi-asserted-by":"publisher","first-page":"1019","DOI":"10.1051\/m2an:2005044","volume":"39","author":"L Dede\u2019","year":"2005","unstructured":"Dede\u2019, L., Quarteroni, A.: Optimal control and numerical adaptivity for advection-diffusion equations. M2AN Math. Model. Numer. Anal. 39(5), 1019\u20131040 (2005)","journal-title":"M2AN Math. Model. Numer. Anal."},{"issue":"1","key":"1087_CR12","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1007\/s00222-009-0196-4","volume":"178","author":"Ferreira D Dos Santos","year":"2009","unstructured":"Dos Santos, Ferreira D., Kenig, C.E., Salo, M., Uhlmann, G.: Limiting Carleman weights and anisotropic inverse problems. Invent. Math. 178(1), 119\u2013171 (2009)","journal-title":"Invent. Math."},{"key":"1087_CR13","series-title":"Applied Mathematical Sciences","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4355-5","volume-title":"Theory and Practice of Finite Elements","author":"A Ern","year":"2004","unstructured":"Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)"},{"issue":"1","key":"1087_CR14","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1051\/m2an:2006006","volume":"40","author":"A Ern","year":"2006","unstructured":"Ern, A., Guermond, J.-L.: Evaluation of the condition number in linear systems arising in finite element approximations. M2AN Math. Model. Numer. Anal. 40(1), 29\u201348 (2006)","journal-title":"M2AN Math. Model. Numer. Anal."},{"issue":"3\u20134","key":"1087_CR15","first-page":"251","volume":"20","author":"F Hecht","year":"2012","unstructured":"Hecht, F.: New development in FreeFem++. J. Numer. Math. 20(3\u20134), 251\u2013265 (2012)","journal-title":"J. Numer. Math."},{"issue":"2\u20133","key":"1087_CR16","first-page":"237","volume":"27","author":"M Hinze","year":"2009","unstructured":"Hinze, M., Yan, N., Zhou, Z.: Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math. 27(2\u20133), 237\u2013253 (2009)","journal-title":"J. Comput. Math."},{"issue":"3","key":"1087_CR17","doi-asserted-by":"publisher","first-page":"712","DOI":"10.1051\/cocv\/2011168","volume":"18","author":"J Le Rousseau","year":"2012","unstructured":"Le Rousseau, J., Lebeau, G.: On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM Control Optim. Calc. Var. 18(3), 712\u2013747 (2012)","journal-title":"ESAIM Control Optim. Calc. Var."},{"issue":"1","key":"1087_CR18","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1137\/S0036142997315172","volume":"36","author":"P Monk","year":"1999","unstructured":"Monk, P., S\u00fcli, E.: The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals. SIAM J. Numer. Anal. 36(1), 251\u2013274 (1999)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"1087_CR19","doi-asserted-by":"publisher","first-page":"1157","DOI":"10.1007\/s00208-011-0712-x","volume":"353","author":"E Malinnikova","year":"2012","unstructured":"Malinnikova, E., Vessella, S.: Quantitative uniqueness for elliptic equations with singular lower order terms. Math. Ann. 353(4), 1157\u20131181 (2012)","journal-title":"Math. Ann."},{"issue":"1","key":"1087_CR20","doi-asserted-by":"publisher","first-page":"198","DOI":"10.1016\/j.cam.2008.01.006","volume":"223","author":"N Yan","year":"2009","unstructured":"Yan, N., Zhou, Z.: A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation. J. Comput. Appl. Math. 223(1), 198\u2013217 (2009)","journal-title":"J. Comput. Appl. Math."},{"key":"1087_CR21","series-title":"Graduate Studies in Mathematics","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/138","volume-title":"Semiclassical Analysis","author":"M Zworski","year":"2012","unstructured":"Zworski, M.: Semiclassical Analysis. Graduate Studies in Mathematics, vol. 138. American Mathematical Society, Providence, RI (2012)"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-019-01087-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-019-01087-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-019-01087-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,12,13]],"date-time":"2020-12-13T00:39:30Z","timestamp":1607819970000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-019-01087-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,12,14]]},"references-count":21,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,3]]}},"alternative-id":["1087"],"URL":"https:\/\/doi.org\/10.1007\/s00211-019-01087-x","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,12,14]]},"assertion":[{"value":"10 November 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 July 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 December 2019","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}