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Numer. Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>In this paper, we study a new numerical method for the solution of partial differential equations on evolving surfaces. The numerical method is built on the stabilized trace finite element method (TraceFEM) for the spatial discretization and finite differences for the time discretization. The TraceFEM uses a stationary background mesh, which can be chosen independent of time and the position of the surface. The stabilization ensures well-conditioning of the algebraic systems and defines a regular extension of the solution from the surface to its volumetric neighborhood. Having such an extension is essential for the numerical method to be well defined. The paper proves numerical stability and optimal order error estimates for the case of simplicial background meshes and finite element spaces of order $m\\geq1$. For the algebraic condition numbers of the resulting systems we prove estimates, which are independent of the position of the interface. The method allows that the surface and its evolution are given implicitly with the help of an indicator function. Results of numerical experiments for a set of 2D evolving surfaces are provided.<\/jats:p>","DOI":"10.1137\/17m1148633","type":"journal-article","created":{"date-parts":[[2018,6,20]],"date-time":"2018-06-20T14:17:31Z","timestamp":1529504251000},"page":"1643-1672","source":"Crossref","is-referenced-by-count":45,"title":["A Stabilized Trace Finite Element Method for Partial Differential Equations on Evolving Surfaces"],"prefix":"10.1137","volume":"56","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0170-8468","authenticated-orcid":true,"given":"Christoph","family":"Lehrenfeld","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maxim A.","family":"Olshanskii","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xianmin","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,6,20]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.4171\/IFB\/338"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2001.6937"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2014.10.044"},{"key":"atypb4","volume-title":"Cut Finite Element Methods for Partial Differential Equations on Embedded Manifolds of Arbitrary Codimensions, preprint, https:\/\/arxiv.org\/abs\/1610.01660","author":"Burman E.","year":"2016"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/S1359-6454(97)00074-8"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/130948641"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/050642873"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drl023"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492913000056"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2012-02601-9"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2007.05.025"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/0723004"},{"key":"atypb14","first-page":"44","volume-title":"Proceedings of ALGORITMY 2016","author":"Engwer C.","year":"2016"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1002\/nme.5121"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/130920095"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/16M1102203"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2015013"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-19686-7"},{"key":"atypb20","volume-title":"Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems","author":"Hairer E.","year":"2002","edition":"2"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(02)00524-8"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2015.05.010"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2016.04.012"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.06.013"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/130943534"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2015.12.005"},{"key":"atypb27","volume-title":"An Eulerian Finite Element Method for PDEs in Time-Dependent Domains, preprint, https:\/\/arxiv.org\/abs\/1803.01779","author":"Lehrenfeld C.","year":"2018"},{"key":"atypb28","volume-title":"A Stabilized Trace Finite Element Method for Partial Differential Equations on Evolving Surfaces, preprint, https:\/\/arxiv.org\/abs\/1709.07117v2","author":"Lehrenfeld C.","year":"2017"},{"key":"atypb29","volume-title":"Analysis of a High Order Unfitted Finite Element Method for Elliptic Interface Problems, preprint, https:\/\/arxiv.org\/abs\/1602.02970","author":"Lehrenfeld C.","year":"2016"},{"key":"atypb30","first-page":"163","volume-title":"ACM","author":"Lorensen W. 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