{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:02:51Z","timestamp":1776844971396,"version":"3.51.2"},"reference-count":8,"publisher":"American Mathematical Society (AMS)","issue":"277","license":[{"start":{"date-parts":[[2012,7,11]],"date-time":"2012-07-11T00:00:00Z","timestamp":1341964800000},"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                    In this paper, we give the first a priori error analysis of the hybridizable discontinuous Galerkin (HDG) methods for Timoshenko beams. The analysis is based on the use of a projection especially designed to fit the structure of the numerical traces of the HDG method. This property allows us to prove in a very concise manner that the projection of the errors is bounded in terms of the distance between the exact solution and its projection. The study of the influence of the stabilization function on the approximation is then reduced to the study of how they affect the approximation properties of the projection in a single element. Surprisingly, and unlike any other discontinuous Galerkin method, this can be done without assuming any positivity property of the stabilization function of the HDG method. We apply this approach to HDG methods using polynomials of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k greater-than-or-equal-to 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k\\ge 0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in all of the unknowns, and show that the projection of the error in each of them superconverges with order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k plus 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k+2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k greater-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k \\ge 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and converges with order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1\">\n                        <mml:semantics>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">1<\/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=\"k equals 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k=0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . As a result, we show that the HDG methods converge with optimal order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k+1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for all the unknowns, and that they are free from shear locking. Finally, we show that all of the numerical traces converge with order\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 k plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2k+1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Numerical experiments validating these results are shown.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2011-02522-6","type":"journal-article","created":{"date-parts":[[2011,7,11]],"date-time":"2011-07-11T10:05:55Z","timestamp":1310378755000},"page":"131-151","source":"Crossref","is-referenced-by-count":17,"title":["A projection-based error analysis of HDG methods for Timoshenko beams"],"prefix":"10.1090","volume":"81","author":[{"given":"Fatih","family":"Celiker","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bernardo","family":"Cockburn","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ke","family":"Shi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2011,7,11]]},"reference":[{"key":"1","unstructured":"F. Celiker, Discontinuous Galerkin methods for Structural Mechanics, Ph.D. thesis, University of Minnesota, 2005."},{"issue":"1-3","key":"2","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s10915-005-9057-5","article-title":"Element-by-element post-processing of discontinuous Galerkin methods for Timoshenko beams","volume":"27","author":"Celiker, Fatih","year":"2006","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"F. Celiker, B. Cockburn, S. G\u00fczey, R. Kannapady, S.-C. Soon, H. K. Stolarski, and K. K. Tamma, Discontinuous Galerkin methods for Timoshenko beams, Numerical Mathematics and Advanced Applications, ENUMATH 2003, Springer, 2003, pp. 221\u2013231.","DOI":"10.1007\/978-3-642-18775-9_19"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-010-9357-2","article-title":"Hybridizable discontinuous Galerkin methods for Timoshenko beams","volume":"44","author":"Celiker, Fatih","year":"2010","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"6","key":"5","doi-asserted-by":"publisher","first-page":"2297","DOI":"10.1137\/050635821","article-title":"Locking-free optimal discontinuous Galerkin methods for Timoshenko beams","volume":"44","author":"Celiker, Fatih","year":"2006","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"1319","DOI":"10.1137\/070706616","article-title":"Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems","volume":"47","author":"Cockburn, Bernardo","year":"2009","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"271","key":"7","doi-asserted-by":"publisher","first-page":"1351","DOI":"10.1090\/S0025-5718-10-02334-3","article-title":"A projection-based error analysis of HDG methods","volume":"79","author":"Cockburn, Bernardo","year":"2010","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","unstructured":"S. P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bar, Philosophical Magazine 41 (1921), 744\u2013746.","DOI":"10.1080\/14786442108636264"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2012-81-277\/S0025-5718-2011-02522-6\/S0025-5718-2011-02522-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2012-81-277\/S0025-5718-2011-02522-6\/S0025-5718-2011-02522-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:01:58Z","timestamp":1776790918000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2012-81-277\/S0025-5718-2011-02522-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,7,11]]},"references-count":8,"journal-issue":{"issue":"277","published-print":{"date-parts":[[2012,1]]}},"alternative-id":["S0025-5718-2011-02522-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02522-6","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":[[2011,7,11]]}}}