{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T11:39:16Z","timestamp":1782301156616,"version":"3.54.5"},"reference-count":31,"publisher":"American Mathematical Society (AMS)","issue":"274","license":[{"start":{"date-parts":[[2011,9,2]],"date-time":"2011-09-02T00:00:00Z","timestamp":1314921600000},"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                    In this paper, we analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the velocity and the pressure converge with the optimal order of convergence of\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                    in\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                    for any\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                    . Moreover, taking advantage of the superconvergence properties of the velocity, we introduce a new element-by-element postprocessing to obtain a new velocity approximation which is exactly divergence-free,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper H left-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold\">H<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbf {H}(<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    div\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <mml:annotation encoding=\"application\/x-tex\">)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -conforming, and converges 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                    for\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 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                    . Numerical experiments are presented which validate the theoretical results.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2010-02410-x","type":"journal-article","created":{"date-parts":[[2010,12,29]],"date-time":"2010-12-29T14:39:21Z","timestamp":1293633561000},"page":"723-760","source":"Crossref","is-referenced-by-count":145,"title":["Analysis of HDG methods for Stokes flow"],"prefix":"10.1090","volume":"80","author":[{"given":"Bernardo","family":"Cockburn","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jayadeep","family":"Gopalakrishnan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ngoc","family":"Nguyen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jaume","family":"Peraire","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Francisco-Javier","family":"Sayas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2010,9,2]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"1749","DOI":"10.1137\/S0036142901384162","article-title":"Unified analysis of discontinuous Galerkin methods for elliptic problems","volume":"39","author":"Arnold, Douglas N.","year":"2001","journal-title":"SIAM J. 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