{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:56:50Z","timestamp":1776841010668,"version":"3.51.2"},"reference-count":26,"publisher":"American Mathematical Society (AMS)","issue":"358","license":[{"start":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T00:00:00Z","timestamp":1770336000000},"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>\n                    A new virtual element method, augmented with local least-squares residuals, on arbitrary polygonal meshes is proposed for the Reissner-Mindlin plate model, where the families of the Lagrangian virtual elements of any equal-order or unequal-order virtual elements\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket upper W Subscript k Baseline right-bracket squared minus upper W Subscript m\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mo>[<\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>W<\/mml:mi>\n                                  <mml:mi>k<\/mml:mi>\n                                <\/mml:msub>\n                                <mml:mo>]<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>W<\/mml:mi>\n                              <mml:mi>m<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\left [W_k\\right ]^2-W_m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are used for approximating the rotation and the transverse displacement while the shear stress is locally computed at the element levels by the local\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                    projections from the discontinuous polynomial elements\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket double-struck upper P Subscript script l Baseline right-bracket squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:msub>\n                                <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                  <mml:mi mathvariant=\"double-struck\">P<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mi>\n                                  \u2113\n                                  \n                                <\/mml:mi>\n                              <\/mml:msub>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">\\left [\\mathbb {P}_\\ell \\right ]^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m minus 1 less-than-or-equal-to script l less-than-or-equal-to min left-parenthesis m plus 1 comma k right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u2113\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mo movablelimits=\"true\" form=\"prefix\">min<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>m<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">m-1\\le \\ell \\le \\min \\left (m+1,k\\right )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Stability and error estimates are established, which hold for all values of the plate thickness (including the zero thickness) and for all the mesh sizes and for all the variables (rotation, transverse displacement and shear stress) in their norms independent of the mesh size and the plate thickness. Numerical experiments are performed to illustrate the method and the theoretical results.\n                  <\/p>","DOI":"10.1090\/mcom\/4065","type":"journal-article","created":{"date-parts":[[2025,2,6]],"date-time":"2025-02-06T14:18:56Z","timestamp":1738851536000},"page":"721-772","source":"Crossref","is-referenced-by-count":0,"title":["A least-squares augmented Lagrangian virtual element method for Reissner-Mindlin plates"],"prefix":"10.1090","volume":"95","author":[{"given":"Huoyuan","family":"Duan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ziliang","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Roger","family":"Tan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2025,2,6]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1016\/j.camwa.2013.05.015","article-title":"Equivalent projectors for virtual element methods","volume":"66","author":"Ahmad, B.","year":"2013","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"key":"3","series-title":"SEMA SIMAI Springer Series","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-95319-5","volume-title":"The virtual element method and its applications","volume":"31","year":"[2022] \\copyright2022","ISBN":"https:\/\/id.crossref.org\/isbn\/9783030953188"},{"issue":"6","key":"4","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"},{"key":"5","unstructured":"D. N. Arnold and R. S. 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