{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:25:42Z","timestamp":1787329542120,"version":"build-2736575974"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>This paper establishes a posteriori error analysis for the Stokes equations discretized by an interior penalty type method using $H(\\mathrm{div})$ finite elements. The a posteriori error estimator is then employed for designing two grid refinement strategies; one is locally based and the other is globally based. The locally based refinement technique is believed to be able to capture local singularities in the numerical solution. The numerical formulations for the Stokes problem make use of $H(\\mathrm{div})$ conforming elements of Raviart\u2013Thomas type. Therefore, the finite element solution features a full satisfaction of the continuity equation (mass conservation). The result of this paper provides a rigorous analysis for the method's reliability and efficiency. In particular, an $H^1$ norm a posteriori error estimator is obtained, together with upper and lower bound estimates. Numerical results are presented to verify the new theory of a posteriori error estimators.<\/jats:p>","DOI":"10.1137\/100783996","type":"journal-article","created":{"date-parts":[[2011,2,1]],"date-time":"2011-02-01T18:07:53Z","timestamp":1296583673000},"page":"131-152","source":"Crossref","is-referenced-by-count":8,"title":["A Posteriori Error Estimation for an Interior Penalty Type Method Employing $H(\\mathrm{div})$ Elements for the Stokes Equations"],"prefix":"10.1137","volume":"33","author":[{"given":"Junping","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yanqiu","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiu","family":"Ye","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,2,1]]},"reference":[{"key":"R1","unstructured":"S. Agmon,\n                      Lectures on Elliptic Boundary Problems\n                      , Van Nostrand, Princeton, NJ, 1965."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/0719052"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389710"},{"key":"R4","doi-asserted-by":"crossref","unstructured":"F. Brezzi and M. Fortin,\n                      Mixed and Hybrid Finite Elements\n                      , Springer-Verlag, New York, 1991.","DOI":"10.1007\/978-1-4612-3172-1"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1002\/num.20467"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/080718413"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-00-01264-3"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1995-1284666-9"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-05-01743-6"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/080713069"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2009.11.024"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2009.05.002"},{"key":"R13","unstructured":"J. S. Howell,\n                      Approximation of generalized Stokes problems using dual-mixed finite elements without enrichment\n                      , Internat. J. Numer. Methods Fluids, accepted, DOI 10.1002\/fld.2356."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994278322"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0363(20000415)32:7<823::AID-FLD991>3.0.CO;2-A"},{"key":"R16","doi-asserted-by":"crossref","unstructured":"P. Raviart and J. Thomas,\n                      A mixed finite element method for 2nd order elliptic problems\n                      , in Mathematical Aspects of Finite Element Methods, I. Galligani and E. Magenes, eds., Lecture Notes in Math. 606, Springer-Verlag, Berlin, 1977, pp. 292\u2013315.","DOI":"10.1007\/BFb0064470"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1990-1011446-7"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.fluid.32.1.93"},{"key":"R19","unstructured":"R. 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