{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T23:06:00Z","timestamp":1767913560050,"version":"3.49.0"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,8,1]],"date-time":"2020-08-01T00:00:00Z","timestamp":1596240000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,8,1]],"date-time":"2020-08-01T00:00:00Z","timestamp":1596240000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000741","name":"University of Warwick","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100000741","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2020,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this article, we analyse the domain mapping method approach to approximate statistical moments of solutions to linear elliptic partial differential equations posed over random geometries including smooth surfaces and bulk-surface systems. In particular, we present the necessary geometric analysis required by the domain mapping method to reformulate elliptic equations on random surfaces onto a fixed deterministic surface using a prescribed stochastic parametrisation of the random domain. An abstract analysis of a finite element discretisation coupled with a Monte-Carlo sampling is presented for the resulting elliptic equations with random coefficients posed over the fixed curved reference domain and optimal error estimates are derived. The results from the abstract framework are applied to a model elliptic problem on a random surface and a coupled elliptic bulk-surface system and the theoretical convergence rates are confirmed by numerical experiments.\n<\/jats:p>","DOI":"10.1007\/s00211-020-01139-7","type":"journal-article","created":{"date-parts":[[2020,8,1]],"date-time":"2020-08-01T18:02:46Z","timestamp":1596304966000},"page":"1-49","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["A domain mapping approach for elliptic equations posed on random bulk and surface domains"],"prefix":"10.1007","volume":"146","author":[{"given":"Lewis","family":"Church","sequence":"first","affiliation":[]},{"given":"Ana","family":"Djurdjevac","sequence":"additional","affiliation":[]},{"given":"Charles M.","family":"Elliott","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,8,1]]},"reference":[{"key":"1139_CR1","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-25983-8","volume-title":"Spherical Harmonics and Approximations on the Unit Sphere: An Introduction","author":"K Atkinson","year":"2012","unstructured":"Atkinson, K., Han, W.: Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Lecture Notes in Mathematics, vol. 2044. Springer, Berlin (2012)"},{"issue":"2","key":"1139_CR2","doi-asserted-by":"publisher","first-page":"800","DOI":"10.1137\/S0036142902418680","volume":"42","author":"I Babuska","year":"2004","unstructured":"Babuska, I., Tempone, R., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800\u2013825 (2004)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2\u20133","key":"1139_CR3","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/s00607-008-0003-x","volume":"82","author":"P Bastian","year":"2008","unstructured":"Bastian, P., Blatt, M., Dedner, A., Engwer, C., Kl\u00f6fkorn, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework. Computing 82(2\u20133), 103\u2013119 (2008)","journal-title":"Computing"},{"issue":"2","key":"1139_CR4","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1007\/s00211-007-0086-x","volume":"107","author":"C Canuto","year":"2007","unstructured":"Canuto, C., Kozubek, T.: A fictitious domain approach to the numerical solution of pdes in stochastic domains. Numer. Math. 107(2), 257 (2007)","journal-title":"Numer. Math."},{"issue":"6","key":"1139_CR5","doi-asserted-by":"publisher","first-page":"1173","DOI":"10.1016\/j.camwa.2016.01.005","volume":"71","author":"J Castrillon-Candas","year":"2016","unstructured":"Castrillon-Candas, J., Nobile, F., Tempone, R.: Analytic regularity and collocation approximation for elliptic PDEs with random domain deformations. Comput. Math. Appl. 71(6), 1173\u20131197 (2016)","journal-title":"Comput. Math. Appl."},{"issue":"1","key":"1139_CR6","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s00791-011-0160-x","volume":"14","author":"KA Cliffe","year":"2011","unstructured":"Cliffe, K.A., Giles, M.B., Scheichl, R., Teckentrup, A.L.: Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci. 14(1), 3\u201315 (2011)","journal-title":"Comput. Vis. Sci."},{"issue":"4","key":"1139_CR7","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1615\/Int.J.UncertaintyQuantification.2017019550","volume":"7","author":"M Dambrine","year":"2017","unstructured":"Dambrine, M., Harbrecht, H., Peters, M., Puig, B.: On Bernoulli\u2019s free boundary problem with a random boundary. Int. J. Uncertain. Quantif. 7(4), 335\u2013353 (2017)","journal-title":"Int. J. Uncertain. Quantif."},{"issue":"2","key":"1139_CR8","doi-asserted-by":"publisher","first-page":"921","DOI":"10.1137\/140998652","volume":"54","author":"M Dambrine","year":"2016","unstructured":"Dambrine, M., Greff, I., Harbrecht, H., Puig, B.: Numerical solution of the poisson equation on domains with a thin layer of random thickness. SIAM J. Numer. Anal. 54(2), 921\u2013941 (2016)","journal-title":"SIAM J. Numer. Anal."},{"key":"1139_CR9","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1017\/S0962492904000224","volume":"14","author":"K Deckelnick","year":"2005","unstructured":"Deckelnick, K., Dziuk, G., Elliott, C.M.: Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139\u2013232 (2005)","journal-title":"Acta Numer."},{"issue":"3\u20134","key":"1139_CR10","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1007\/s00607-010-0110-3","volume":"90","author":"A Dedner","year":"2010","unstructured":"Dedner, A., Kl\u00f6fkorn, R., Nolte, M., Ohlberger, M.: A generic interface for parallel and adaptive discretization schemes: abstraction principles and the dune-fem module. Computing 90(3\u20134), 165\u2013196 (2010)","journal-title":"Computing"},{"key":"1139_CR11","unstructured":"Djurdjevac, A.: Random moving domain (2018). arXiv:1808.06970"},{"issue":"4","key":"1139_CR12","doi-asserted-by":"publisher","first-page":"1656","DOI":"10.1137\/17M1149547","volume":"6","author":"A Djurdjevac","year":"2018","unstructured":"Djurdjevac, A., Elliott, C., Kornhuber, R., Ranner, T.: Evolving surface finite element methods for random advection\u2013diffusion equations. SIAM\/ASA J. Uncertain. Quantif. 6(4), 1656\u20131684 (2018)","journal-title":"SIAM\/ASA J. Uncertain. Quantif."},{"key":"1139_CR13","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1017\/S0962492913000056","volume":"22","author":"G Dziuk","year":"2013","unstructured":"Dziuk, G., Elliott, C.: Finite element methods for surface pdes. Acta Numer. 22, 289\u2013396 (2013)","journal-title":"Acta Numer."},{"issue":"2","key":"1139_CR14","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1093\/imanum\/drs022","volume":"33","author":"C Elliott","year":"2012","unstructured":"Elliott, C., Ranner, T.: Finite element analysis for a coupled bulk-surface partial differential equation. IMA J. Numer. Anal. 33(2), 377\u2013402 (2012)","journal-title":"IMA J. Numer. Anal."},{"key":"1139_CR15","unstructured":"Elliott, C., Ranner, T.: A unified theory for continuous in time evolving finite element space approximations to partial differential equations in evolving domains (2017). arXiv:1703.04679"},{"key":"1139_CR16","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511530005","volume-title":"Geometric Applications of Fourier Series and Spherical Harmonics","author":"H Groemer","year":"1996","unstructured":"Groemer, H.: Geometric Applications of Fourier Series and Spherical Harmonics, vol. 61. Cambridge University Press, Cambridge (1996)"},{"key":"1139_CR17","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1017\/S0962492914000075","volume":"23","author":"MD Gunzburger","year":"2014","unstructured":"Gunzburger, M.D., Webster, C.G., Zhang, G.: Stochastic finite element methods for partial differential equations with random input data. Acta Numer. 23, 521\u2013650 (2014)","journal-title":"Acta Numer."},{"issue":"5","key":"1139_CR18","doi-asserted-by":"publisher","first-page":"1533","DOI":"10.1051\/m2an\/2013079","volume":"47","author":"H Harbrecht","year":"2013","unstructured":"Harbrecht, H., Li, J.: First order second moment analysis for stochastic interface problems based on low-rank approximation. ESAIM Math. Model. Numer. Anal. 47(5), 1533\u20131552 (2013)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"issue":"4","key":"1139_CR19","doi-asserted-by":"publisher","first-page":"823","DOI":"10.1007\/s00211-016-0791-4","volume":"134","author":"H Harbrecht","year":"2016","unstructured":"Harbrecht, H., Peters, M., Siebenmorgen, M.: Analysis of the domain mapping method for elliptic diffusion problems on random domains. Numer. Math. 134(4), 823\u2013856 (2016)","journal-title":"Numer. Math."},{"issue":"3","key":"1139_CR20","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1007\/s00211-008-0147-9","volume":"109","author":"H Harbrecht","year":"2008","unstructured":"Harbrecht, H., Schneider, R., Schwab, C.: Sparse second moment analysis for elliptic problems in stochastic domains. Numer. Math. 109(3), 385\u2013414 (2008)","journal-title":"Numer. Math."},{"issue":"6","key":"1139_CR21","doi-asserted-by":"publisher","first-page":"3351","DOI":"10.1137\/110845537","volume":"50","author":"FY Kuo","year":"2012","unstructured":"Kuo, F.Y., Schwab, C., Sloan, I.H.: Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50(6), 3351\u20133374 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"1139_CR22","volume-title":"Linear and quasilinear elliptic equations","author":"OA Ladyzhenskaya","year":"1968","unstructured":"Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and quasilinear elliptic equations. Academic Press, Cambridge (1968)"},{"key":"1139_CR23","series-title":"Scientific Computation","doi-asserted-by":"publisher","DOI":"10.1007\/978-90-481-3520-2","volume-title":"Spectral Methods for Uncertainty Quantification: with Applications to Computational Fluid Dynamics","author":"O Le Ma\u00eetre","year":"2010","unstructured":"Le Ma\u00eetre, O., Knio, O.M.: Spectral Methods for Uncertainty Quantification: with Applications to Computational Fluid Dynamics. Scientific Computation. Springer, Berlin (2010)"},{"key":"1139_CR24","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139017329","volume-title":"An Introduction to Computational Stochastic PDEs","author":"GJ Lord","year":"2014","unstructured":"Lord, G.J., Powell, C.E., Shardlow, T.: An Introduction to Computational Stochastic PDEs, vol. 50. Cambridge University Press, Cambridge (2014)"},{"issue":"12\u201316","key":"1139_CR25","doi-asserted-by":"publisher","first-page":"1295","DOI":"10.1016\/j.cma.2004.05.027","volume":"194","author":"HG Matthies","year":"2005","unstructured":"Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194(12\u201316), 1295\u20131331 (2005)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"45\u201346","key":"1139_CR26","doi-asserted-by":"publisher","first-page":"3066","DOI":"10.1016\/j.cma.2011.07.002","volume":"200","author":"A Nouy","year":"2011","unstructured":"Nouy, A., Chevreuil, M., Safatly, E.: Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains. Comput. Methods Appl. Mech. Eng. 200(45\u201346), 3066\u20133082 (2011)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"51\u201352","key":"1139_CR27","doi-asserted-by":"publisher","first-page":"4663","DOI":"10.1016\/j.cma.2008.06.010","volume":"197","author":"A Nouy","year":"2008","unstructured":"Nouy, A., Clement, A., Schoefs, F., Mo\u00ebs, N.: An extended stochastic finite element method for solving stochastic partial differential equations on random domains. Comput. Methods Appl. Mech. Eng. 197(51\u201352), 4663\u20134682 (2008)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1139_CR28","volume-title":"Methods of Modern Mathematical Physics I: Functional Analysis","author":"M Reed","year":"1972","unstructured":"Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, Cambridge (1972)"},{"key":"1139_CR29","volume-title":"An Analysis of the Finite Element Method","author":"G Strang","year":"1973","unstructured":"Strang, G., Fix, G.J.: An Analysis of the Finite Element Method, vol. 212. Prentice-Hall, Englewood Cliffs (1973)"},{"issue":"1","key":"1139_CR30","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1090\/S0002-9947-1934-1501735-3","volume":"36","author":"H Whitney","year":"1934","unstructured":"Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36(1), 63\u201389 (1934)","journal-title":"Trans. Am. Math. Soc."},{"issue":"3","key":"1139_CR31","doi-asserted-by":"publisher","first-page":"1167","DOI":"10.1137\/040613160","volume":"28","author":"D Xiu","year":"2006","unstructured":"Xiu, D., Tartakovsky, D.M.: Numerical methods for differential equations in random domains. SIAM J. Sci. Comput. 28(3), 1167\u20131185 (2006)","journal-title":"SIAM J. Sci. Comput."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-020-01139-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-020-01139-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-020-01139-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,31]],"date-time":"2021-07-31T23:39:38Z","timestamp":1627774778000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-020-01139-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,1]]},"references-count":31,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,9]]}},"alternative-id":["1139"],"URL":"https:\/\/doi.org\/10.1007\/s00211-020-01139-7","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,8,1]]},"assertion":[{"value":"2 May 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 April 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 August 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}