{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:42:32Z","timestamp":1776789752380,"version":"3.51.2"},"reference-count":34,"publisher":"American Mathematical Society (AMS)","issue":"266","license":[{"start":{"date-parts":[[2009,8,1]],"date-time":"2009-08-01T00:00:00Z","timestamp":1249084800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>The present paper is made up of two parts. In the first part, we study the mathematical stability and convergence of the quadrilateral MITC elements for the Reissner-Mindlin plate problem in an abstract setting. We generalize the Brezzi-Bathe-Fortin conditions to the quadrilateral MITC elements by weakening the second and fourth conditions. Under these conditions, we show the well-posedness of the discrete problem and establish an abstract error estimate in the energy norm. The conclusion of this part is sparsity in the mathematical research of the quadrilateral MITC elements in the sense that one only needs to check these five conditions.<\/p>\n                  <p>\n                    In the second part, we extend four families of rectangular MITC elements of Stenberg and S\u00fcri to the quadrilateral meshes. We prove that these quadrilateral elements satisfy the generalized Brezzi-Bathe-Fortin conditions from the first part. We develop the h-p error estimates in both energy and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    norm for these quadrilateral elements. For the first three families of quadrilateral elements, the error estimates indicate that their convergent rates in both energy and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    norm depend on the mesh distortion parameter\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b1\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We can get optimal error estimates for them provided that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha =1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In addition, we show the optimal convergence rates in energy norm uniformly in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b1\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for the fourth family of quadrilateral elements. Like their rectangular counterparts, these quadrilateral elements are locking-free.\n                  <\/p>","DOI":"10.1090\/s0025-5718-08-02153-4","type":"journal-article","created":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T13:09:23Z","timestamp":1259672963000},"page":"673-711","source":"Crossref","is-referenced-by-count":9,"title":["Analysis for quadrilateral MITC elements for the Reissner-Mindlin plate problem"],"prefix":"10.1090","volume":"78","author":[{"given":"Jun","family":"Hu","sequence":"first","affiliation":[]},{"given":"Zhong-Ci","family":"Shi","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,8,1]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"issue":"6","key":"2","doi-asserted-by":"publisher","first-page":"1276","DOI":"10.1137\/0726074","article-title":"A uniformly accurate finite element method for the Reissner-Mindlin plate","volume":"26","author":"Arnold, Douglas N.","year":"1989","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"239","key":"3","doi-asserted-by":"publisher","first-page":"909","DOI":"10.1090\/S0025-5718-02-01439-4","article-title":"Approximation by quadrilateral finite elements","volume":"71","author":"Arnold, Douglas N.","year":"2002","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","unstructured":"D. N. Arnold, D. Boffi and R. S. Falk. Remarks on quadrilateral Reissner-Mindlin plate elements, in Proceedings of Fifth World Congress on Computational Mechanics, H. A. Mang, F. G. Rammerstorfer and J. Eberhardsteiner, eds."},{"issue":"6","key":"5","doi-asserted-by":"publisher","first-page":"2429","DOI":"10.1137\/S0036142903431924","article-title":"Quadrilateral \ud835\udc3b(\ud835\udc51\ud835\udc56\ud835\udc63) finite elements","volume":"42","author":"Arnold, Douglas N.","year":"2005","journal-title":"SIAM J. Numer. 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Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"750","DOI":"10.1137\/0724049","article-title":"The optimal convergence rate of the \ud835\udc5d-version of the finite element method","volume":"24","author":"Babu\u0161ka, I.","year":"1987","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"9","unstructured":"K. J. Bathe, F. Brezzi and M. Fortin. A simplified analysis of two-plate elements: The MITC4 and MITC9 element, G. N. Pande and J. Middleton (eds.), Numeta 87 Vol.1, Numerical Techniques for Engineering Analysis and Design, Martinus Nijhoff, Amsterdam."},{"key":"10","doi-asserted-by":"crossref","unstructured":"K. J. Bathe and E. Dvorkin. A four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation, Internat. J. Numer. 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