{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,22]],"date-time":"2026-03-22T01:05:55Z","timestamp":1774141555354,"version":"3.50.1"},"reference-count":27,"publisher":"Walter de Gruyter GmbH","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,4,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this article, a priori and a posteriori estimates of conforming and expanded mixed finite element methods for a Kirchhoff equation\nof elliptic type are derived. For the expanded mixed finite element method, a variant of Brouwer\u2019s fixed point argument combined with a monotonicity argument yields the well-posedness of the discrete nonlinear system. Further, a use of both Helmholtz decomposition of <jats:inline-formula id=\"j_cmam-2016-0041_ineq_9999_w2aab3b7e1085b1b6b1aab1c13b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>L<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:tex-math>$L^{2}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-vector valued functions and the discrete Helmholtz decomposition of the Raviart\u2013Thomas finite elements helps in a crucial way to achieve optimal a priori as well as a posteriori\nerror bounds. For both conforming and expanded mixed form, reliable and efficient a posteriori\nestimators are established. Finally, the numerical experiments\nare performed to validate the theoretical convergence rates.<\/jats:p>","DOI":"10.1515\/cmam-2016-0041","type":"journal-article","created":{"date-parts":[[2017,1,10]],"date-time":"2017-01-10T10:01:27Z","timestamp":1484042487000},"page":"217-236","source":"Crossref","is-referenced-by-count":10,"title":["A Priori and A Posteriori Estimates of Conforming and Mixed FEM for a Kirchhoff Equation of Elliptic Type"],"prefix":"10.1515","volume":"17","author":[{"given":"Asha K.","family":"Dond","sequence":"first","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India"}]},{"given":"Amiya K.","family":"Pani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India"}]}],"member":"374","published-online":{"date-parts":[[2017,1,10]]},"reference":[{"key":"2023033114222325807_j_cmam-2016-0041_ref_001_w2aab3b7e1085b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"Ainsworth M. and Oden J. 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