{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,18]],"date-time":"2026-09-18T09:20:39Z","timestamp":1789723239590,"version":"4.0.1"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We introduce a new finite element method, the composite mini element, for the mixed discretization of the Stokes equations on two- and three-dimensional domains that may contain a huge number of geometric details. In standard finite element discretizations of the Stokes problem, such as the classical mini element, the approximation quality is determined by the maximal mesh size of the underlying triangulation, while the computational effort is determined by its number of elements. If the physical domain is very complicated, then the minimal number of simplices, which are necessary to resolve the domain, can be very large and distributed in a nonoptimal way with respect to the approximation quality. In contrast to that, the minimal dimension of the composite mini element space is independent of the number of geometric details. Instead of a geometric resolution of the domain and the boundary condition by the finite element mesh the shape of the finite element functions is adapted to the geometric details. This approach allows low-dimensional approximations even for problems with complicated geometric details such as holes or rough boundaries. We prove its linear (optimal order) approximability and its inf-sup stability. Further, we will be able to control the nonconformity in the space without increasing the space dimension in such a way that the a priori error estimate $\\|{\\mathbf{u}-\\mathbf{u}^{\\mathrm{CME}}}\\|_{1,\\Omega}+\\|{p-p^{\\mathrm{CME}}} \\|_{0,\\Omega}\\lesssim h\\|{\\mathbf{f}}\\|_{0,\\Omega}$ holds. Thereby, in contrast to the classical methods, the choice of the mesh size parameter h is not constrained by the size of geometric details. In addition, it turns out that the method can be viewed as a coarse-scale generalization of the classical mini element approach; i.e., it reduces the computational effort, while the approximation quality depends on the (coarse) mesh size in the usual way.<\/jats:p>","DOI":"10.1137\/070704356","type":"journal-article","created":{"date-parts":[[2008,9,18]],"date-time":"2008-09-18T10:32:23Z","timestamp":1221733943000},"page":"3181-3206","source":"Crossref","is-referenced-by-count":20,"title":["The Composite Mini Element\u2014Coarse Mesh Computation of Stokes Flows on Complicated Domains"],"prefix":"10.1137","volume":"46","author":[{"given":"Daniel","family":"Peterseim","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefan A.","family":"Sauter","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,9,17]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050181"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/4.3.309"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"S. Brenner and L. R. Scott,\n                      The Mathematical Theory of Finite Element Methods\n                      , Springer-Verlag, New York, 1994.","DOI":"10.1007\/978-1-4757-4338-8"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"F. Brezzi and M. Fortin,\n                      Mixed and Hybrid Finite Element Methods\n                      , Springer-Verlag, New York, 1991.","DOI":"10.1007\/978-1-4612-3172-1"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\n                      The Finite Element Method for Elliptic Problems\n                      , North\u2013Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1099-1476(19960725)19:11<857::AID-MMA798>3.0.CO;2-D"},{"key":"R7","first-page":"77","volume":"9","author":"Cl\u00e9ment Ph.","year":"1975","journal-title":"RAIRO Anal. Num\u00e9r.","ISSN":"https:\/\/id.crossref.org\/issn\/0399-0516","issn-type":"print"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/0522040"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"G. Duvaut and J. L. Lions,\n                      Inequalities in Mechanics and Physics\n                      , Springer-Verlag, Berlin, Heidelberg, New York, 1976.","DOI":"10.1007\/978-3-642-66165-5"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"V. Girault and P.A. Raviart,\n                      Finite Element Approximation of the Navier-Stokes Equations\n                      , Springer-Verlag, Berlin, Heidelberg, New York, 1979.","DOI":"10.1007\/BFb0063447"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"W. Hackbusch and S. A. Sauter,\n                      Composite finite elements for problems containing small geometric details. Part\n                      II:\n                      Implementation and numerical results\n                      , Comput. Vis. Sci., 1 (1997) pp. 15\u201325.","DOI":"10.1007\/s007910050002"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050248"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"R. Kornhuber and H. Yserentant,\n                      Multilevel methods for elliptic problems on domains not resolved by the coarse grid\n                      , in Domain Decomposition Methods in Scientific and Engineering Computing, Contemp. Math. 180, AMS, Providence, RI, 1994, pp. 49\u201360.","DOI":"10.1090\/conm\/180\/01956"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/1981150302371"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"D. Peterseim,\n                      The Composite Mini Element: A Mixed FEM for the Stokes Equations on Complicated Domains\n                      , Ph.D. thesis, Universit\u00e4t Z\u00fcrich, Z\u00fcrich, Switzerland, 2007.","DOI":"10.1002\/pamm.200700691"},{"key":"R16","unstructured":"M. Rech,\n                      Composite Finite Elements: An Adaptive Two-Scale Approach to the Nonconforming Approximation of Dirichlet Problems on Complicated Domains\n                      , Ph.D. thesis, Universit\u00e4t Z\u00fcrich, Z\u00fcrich, Switzerland, 2006."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-005-0654-x"},{"key":"R18","first-page":"61","volume":"7","author":"Sauter S. A.","year":"1999","journal-title":"East-West J. Numer. Math."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1990-1011446-7"},{"key":"R20","unstructured":"E. M. 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