{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:47:53Z","timestamp":1776728873802,"version":"3.51.2"},"reference-count":7,"publisher":"American Mathematical Society (AMS)","issue":"244","license":[{"start":{"date-parts":[[2004,3,4]],"date-time":"2004-03-04T00:00:00Z","timestamp":1078358400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We consider a bilinear reduced-strain finite element of the MITC family for a shallow Reissner-Naghdi type shell. We estimate the consistency error of the element in both membrane- and bending-dominated states of deformation. We prove that in the membrane-dominated case, under severe assumptions on the domain, the finite element mesh and the regularity of the solution, an error bound\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis h plus t Superscript negative 1 Baseline h Superscript 1 plus s Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mi>h<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mi>s<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(h + t^{-1}h^{1+s})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    can be obtained if the contribution of transverse shear is neglected. Here\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t\">\n                        <mml:semantics>\n                          <mml:mi>t<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the thickness of the shell,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h\">\n                        <mml:semantics>\n                          <mml:mi>h<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    the mesh spacing, and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s\">\n                        <mml:semantics>\n                          <mml:mi>s<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">s<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    a smoothness parameter. In the bending-dominated case, the uniformly optimal bound\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis h right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(h)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is achievable but requires that membrane and transverse shear strains are of order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis t squared right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>t<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(t^2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"t right-arrow 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">t \\rightarrow 0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In this case we also show that under sufficient regularity assumptions the asymptotic consistency error has the bound\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper O left-parenthesis h right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>O<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">O(h)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-03-01508-4","type":"journal-article","created":{"date-parts":[[2003,6,20]],"date-time":"2003-06-20T10:23:27Z","timestamp":1056104607000},"page":"1635-1653","source":"Crossref","is-referenced-by-count":9,"title":["Analysis of a bilinear finite element for shallow shells. II: Consistency error"],"prefix":"10.1090","volume":"72","author":[{"given":"Ville","family":"Havu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juhani","family":"Pitk\u00e4ranta","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2003,3,4]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"K.J. Bathe, E.N. Dvorkin, A formulation of general shell elements \u2013 the use of mixed interpolation of tensorial components, Int. J. Numer. Methods Engrg. 22 (1986) 697-722.","DOI":"10.1002\/nme.1620220312"},{"key":"2","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4338-8","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0387941932"},{"key":"3","doi-asserted-by":"crossref","unstructured":"V. Havu, J. Pitk\u00e4ranta, Analysis of a bilinear finite element for shallow shells I: Approximation of inextensional deformations, Math. Comp. 71 (2002) 923-943.","DOI":"10.1090\/S0025-5718-01-01376-X"},{"key":"4","doi-asserted-by":"crossref","unstructured":"M. Malinen, On the classical shell model underlying bilinear degenerated shell finite elements, Int. J. Numer. Methods Engrg. 52 (2001) 389-416.","DOI":"10.1002\/nme.216"},{"key":"5","doi-asserted-by":"publisher","first-page":"549","DOI":"10.2307\/1968939","article-title":"Transformationen von algebraischem Typ","volume":"40","author":"Kober, Hermann","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"11-12","key":"6","doi-asserted-by":"publisher","first-page":"1323","DOI":"10.1016\/S0045-7825(00)00163-8","article-title":"The first locking-free plane-elastic finite element: historia mathematica","volume":"190","author":"Pitk\u00e4ranta, Juhani","year":"2000","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/BF01385524","article-title":"The problem of membrane locking in finite element analysis of cylindrical shells","volume":"61","author":"Pitk\u00e4ranta, Juhani","year":"1992","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2003-72-244\/S0025-5718-03-01508-4\/S0025-5718-03-01508-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-244\/S0025-5718-03-01508-4\/S0025-5718-03-01508-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:25:18Z","timestamp":1776727518000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-244\/S0025-5718-03-01508-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,3,4]]},"references-count":7,"journal-issue":{"issue":"244","published-print":{"date-parts":[[2003,10]]}},"alternative-id":["S0025-5718-03-01508-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-03-01508-4","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2003,3,4]]}}}