{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T04:24:01Z","timestamp":1777091041323,"version":"3.51.4"},"reference-count":63,"publisher":"American Mathematical Society (AMS)","issue":"285","license":[{"start":{"date-parts":[[2014,7,16]],"date-time":"2014-07-16T00:00:00Z","timestamp":1405468800000},"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                    We present the first a priori error analysis of the hybridizable discontinuous Galerkin methods for the acoustic equation in the time-continuous case. We show that the velocity and the gradient converge with the optimal order 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 the\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 uniformly in time whenever 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 \\geq 0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are used. Finally, we show how to take advantage of this local postprocessing to obtain an approximation to the original scalar unknown also converging 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 \\geq 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This puts on firm mathematical ground the numerical results obtained in J. Comput. Phys.\n                    <bold>230<\/bold>\n                    (2011), 3695\u20133718.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2013-02743-3","type":"journal-article","created":{"date-parts":[[2013,7,16]],"date-time":"2013-07-16T13:46:54Z","timestamp":1373982414000},"page":"65-85","source":"Crossref","is-referenced-by-count":46,"title":["Uniform-in-time superconvergence of the HDG methods for the acoustic wave equation"],"prefix":"10.1090","volume":"83","author":[{"given":"Bernardo","family":"Cockburn","sequence":"first","affiliation":[]},{"given":"Vincent","family":"Quenneville-B\u00e9lair","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2013,7,16]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"R. M. Alford, K. R. Kelly, and D. M. Boore, Accuracy of finite-difference modeling of the acoustic wave equation, Geophysics 39 (1974), no. 6, 834\u2013842.","DOI":"10.1190\/1.1440470"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"564","DOI":"10.1137\/0713048","article-title":"Error estimates for finite element methods for second order hyperbolic equations","volume":"13","author":"Baker, Garth A.","year":"1976","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"10","key":"3","doi-asserted-by":"publisher","first-page":"1043","DOI":"10.1002\/fld.562","article-title":"Superconvergence and \ud835\udc3b(\ud835\udc51\ud835\udc56\ud835\udc63) projection for discontinuous Galerkin methods","volume":"42","author":"Bastian, Peter","year":"2003","journal-title":"Internat. J. Numer. Methods Fluids","ISSN":"https:\/\/id.crossref.org\/issn\/0271-2091","issn-type":"print"},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"1053","DOI":"10.1137\/S0036142998345499","article-title":"An analysis of new mixed finite elements for the approximation of wave propagation problems","volume":"37","author":"B\u00e9cache, E.","year":"2000","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"5","first-page":"319","article-title":"New high-order integral methods in computational electromagnetism","volume":"5","author":"Bruno, Oscar P.","year":"2004","journal-title":"CMES Comput. Model. Eng. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/1526-1492","issn-type":"print"},{"issue":"4","key":"6","doi-asserted-by":"publisher","first-page":"530","DOI":"10.1080\/17455030.2010.499151","article-title":"Efficient high-order evaluation of scattering by periodic surfaces: vector-parametric gratings and geometric singularities","volume":"20","author":"Bruno, Oscar P.","year":"2010","journal-title":"Waves Random Complex Media","ISSN":"https:\/\/id.crossref.org\/issn\/1745-5030","issn-type":"print"},{"issue":"277","key":"7","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1090\/S0025-5718-2011-02522-6","article-title":"A projection-based error analysis of HDG methods for Timoshenko beams","volume":"81","author":"Celiker, Fatih","year":"2012","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"8","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":"277","key":"9","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1090\/S0025-5718-2011-02525-1","article-title":"Uniform-in-time superconvergence of HDG methods for the heat equation","volume":"81","author":"Chabaud, Brandon","year":"2012","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"10","doi-asserted-by":"publisher","first-page":"1267","DOI":"10.1093\/imanum\/drr058","article-title":"Analysis of variable-degree HDG methods for convection-diffusion equations. Part I: general nonconforming meshes","volume":"32","author":"Chen, Yanlai","year":"2012","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"6","key":"11","doi-asserted-by":"publisher","first-page":"4044","DOI":"10.1137\/090747701","article-title":"Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convection-diffusion equations in one space dimension","volume":"47","author":"Cheng, Yingda","year":"2010","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"279","key":"12","doi-asserted-by":"publisher","first-page":"1355","DOI":"10.1090\/S0025-5718-2011-02575-5","article-title":"An analysis of HDG methods for the vorticity-velocity-pressure formulation of the Stokes problem in three dimensions","volume":"81","author":"Cockburn, Bernardo","year":"2012","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"264","key":"13","doi-asserted-by":"publisher","first-page":"1887","DOI":"10.1090\/S0025-5718-08-02123-6","article-title":"A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems","volume":"77","author":"Cockburn, Bernardo","year":"2008","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1-3","key":"14","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1007\/s10915-009-9279-z","article-title":"A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems","volume":"40","author":"Cockburn, Bernardo","year":"2009","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"2","key":"15","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":"274","key":"16","doi-asserted-by":"publisher","first-page":"723","DOI":"10.1090\/S0025-5718-2010-02410-X","article-title":"Analysis of HDG methods for Stokes flow","volume":"80","author":"Cockburn, Bernardo","year":"2011","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"271","key":"17","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"},{"issue":"265","key":"18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0025-5718-08-02146-7","article-title":"Superconvergent discontinuous Galerkin methods for second-order elliptic problems","volume":"78","author":"Cockburn, Bernardo","year":"2009","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"251","key":"19","doi-asserted-by":"publisher","first-page":"1067","DOI":"10.1090\/S0025-5718-04-01718-1","article-title":"A locally conservative LDG method for the incompressible Navier-Stokes equations","volume":"74","author":"Cockburn, Bernardo","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1-3","key":"20","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1007\/s10915-010-9359-0","article-title":"A comparison of HDG methods for Stokes flow","volume":"45","author":"Cockburn, B.","year":"2010","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"279","key":"21","doi-asserted-by":"publisher","first-page":"1327","DOI":"10.1090\/S0025-5718-2011-02550-0","article-title":"Conditions for superconvergence of HDG methods for second-order elliptic problems","volume":"81","author":"Cockburn, Bernardo","year":"2012","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"22","unstructured":"B. Cockburn and F.J. Sayas, Divergence-conforming HDG methods for Stokes flow, submitted."},{"issue":"282","key":"23","doi-asserted-by":"publisher","first-page":"651","DOI":"10.1090\/S0025-5718-2012-02644-5","article-title":"Conditions for superconvergence of HDG methods for Stokes flow","volume":"82","author":"Cockburn, Bernardo","year":"2013","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"13-16","key":"24","doi-asserted-by":"publisher","first-page":"1305","DOI":"10.1016\/j.cma.2007.11.006","article-title":"Finite element approximation of the hyperbolic wave equation in mixed form","volume":"197","author":"Codina, Ramon","year":"2008","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"4","key":"25","doi-asserted-by":"publisher","first-page":"1266","DOI":"10.1137\/S0036142993246445","article-title":"Construction analysis of fourth-order finite difference schemes for the acoustic wave equation in nonhomogeneous media","volume":"33","author":"Cohen, Gary","year":"1996","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"26","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03537-5","volume-title":"Inverse acoustic and electromagnetic scattering theory","volume":"93","author":"Colton, David","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/354062838X","edition":"2"},{"issue":"1-3","key":"27","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/0045-7825(90)90165-I","article-title":"A priori estimates for mixed finite element methods for the wave equation","volume":"82","author":"Cowsar, Lawrence C.","year":"1990","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"2","key":"28","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1137\/0733026","article-title":"A priori estimates for mixed finite element approximations of second-order hyperbolic equations with absorbing boundary conditions","volume":"33","author":"Cowsar, Lawrence C.","year":"1996","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"29","doi-asserted-by":"crossref","unstructured":"M. A. Dablain, The application of high-order differencing to the scalar wave equation, Geophysics 51 (1986), no. 1, 54\u201366.","DOI":"10.1190\/1.1442040"},{"key":"30","doi-asserted-by":"publisher","first-page":"880","DOI":"10.1137\/0710073","article-title":"\ud835\udc3f\u00b2-estimates for Galerkin methods for second order hyperbolic equations","volume":"10","author":"Dupont, Todd","year":"1973","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"31","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4355-5","volume-title":"Theory and practice of finite elements","volume":"159","author":"Ern, Alexandre","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0387205748"},{"key":"32","series-title":"Graduate Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/019","volume-title":"Partial differential equations","volume":"19","author":"Evans, Lawrence C.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0821807722"},{"issue":"6","key":"33","doi-asserted-by":"publisher","first-page":"1886","DOI":"10.1016\/j.cam.2009.08.094","article-title":"Local time stepping and discontinuous Galerkin methods for symmetric first order hyperbolic systems","volume":"234","author":"Ezziani, Abdela\u00e2ziz","year":"2010","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"3","key":"34","doi-asserted-by":"publisher","first-page":"935","DOI":"10.1137\/S0036142997329463","article-title":"Explicit finite element methods for symmetric hyperbolic equations","volume":"36","author":"Falk, Richard S.","year":"1999","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"35","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and hybrid finite element methods","volume":"15","author":"Brezzi, Franco","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387975829"},{"issue":"1","key":"36","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1051\/m2an\/1989230101031","article-title":"Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations","volume":"23","author":"Gastaldi, Lucia","year":"1989","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"2","key":"37","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1051\/m2an\/1988220202431","article-title":"On the application of mixed finite element methods to the wave equations","volume":"22","author":"Geveci, Tunc","year":"1988","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"2","key":"38","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1016\/0021-9991(87)90140-9","article-title":"A fast algorithm for particle simulations","volume":"73","author":"Greengard, L.","year":"1987","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"6","key":"39","doi-asserted-by":"publisher","first-page":"2408","DOI":"10.1137\/05063194X","article-title":"Discontinuous Galerkin finite element method for the wave equation","volume":"44","author":"Grote, Marcus J.","year":"2006","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"40","doi-asserted-by":"publisher","first-page":"420","DOI":"10.1016\/j.cam.2006.08.003","article-title":"Numerical solution of the acoustic wave equation using Raviart-Thomas elements","volume":"206","author":"Jenkins, Eleanor W.","year":"2007","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"5","key":"41","doi-asserted-by":"publisher","first-page":"1698","DOI":"10.1137\/S0036142901388068","article-title":"A priori error estimates for mixed finite element approximations of the acoustic wave equation","volume":"40","author":"Jenkins, Eleanor W.","year":"2002","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"42","doi-asserted-by":"publisher","first-page":"1292","DOI":"10.1137\/S003614290342827X","article-title":"Difference approximations of the Neumann problem for the second order wave equation","volume":"42","author":"Kreiss, Heinz-Otto","year":"2004","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"43","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1142\/S0218396X08003762","article-title":"Computational aspects of the discontinuous Galerkin method for the wave equation","volume":"16","author":"L\u00e4hivaara, Timo","year":"2008","journal-title":"J. Comput. Acoust.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-396X","issn-type":"print"},{"issue":"23","key":"44","doi-asserted-by":"publisher","first-page":"8779","DOI":"10.1016\/j.jcp.2009.08.027","article-title":"A new time-space domain high-order finite-difference method for the acoustic wave equation","volume":"228","author":"Liu, Yang","year":"2009","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"45","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1007\/s10915-004-4132-5","article-title":"A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media","volume":"22\/23","author":"Monk, Peter","year":"2005","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"key":"46","doi-asserted-by":"crossref","unstructured":"N. C. Nguyen, J. Peraire, and B. Cockburn, A hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations (AIAA Paper 2010-362), Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit (Orlando, Florida), January 2010.","DOI":"10.2514\/6.2010-362"},{"issue":"10","key":"47","doi-asserted-by":"publisher","first-page":"3695","DOI":"10.1016\/j.jcp.2011.01.035","article-title":"High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics","volume":"230","author":"Nguyen, N. C.","year":"2011","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"4","key":"48","doi-asserted-by":"publisher","first-page":"1147","DOI":"10.1016\/j.jcp.2010.10.032","article-title":"An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations","volume":"230","author":"Nguyen, N. C.","year":"2011","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"49","doi-asserted-by":"crossref","unstructured":"N. C. Nguyen, J. Peraire, and B. Cockburn, Hybridizable discontinuous Galerkin methods, Proceedings of the International Conference on Spectral and High Order Methods (Trondheim, Norway), Lect. Notes Comput. Sci. Engrg., Springer Verlag, June 2009.","DOI":"10.1007\/978-3-642-15337-2_4"},{"issue":"9","key":"50","doi-asserted-by":"publisher","first-page":"3232","DOI":"10.1016\/j.jcp.2009.01.030","article-title":"An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations","volume":"228","author":"Nguyen, N. C.","year":"2009","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"23","key":"51","doi-asserted-by":"publisher","first-page":"8841","DOI":"10.1016\/j.jcp.2009.08.030","article-title":"An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations","volume":"228","author":"Nguyen, N. C.","year":"2009","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"9-12","key":"52","doi-asserted-by":"publisher","first-page":"582","DOI":"10.1016\/j.cma.2009.10.007","article-title":"A hybridizable discontinuous Galerkin method for Stokes flow","volume":"199","author":"Nguyen, N. C.","year":"2010","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"53","doi-asserted-by":"crossref","unstructured":"N. Nishimura, Fast multipole accelerated boundary integral equation methods, Applied Mechanics Reviews 55 (2002), no. 4, 299\u2013324.","DOI":"10.1115\/1.1482087"},{"key":"54","doi-asserted-by":"crossref","unstructured":"J. Peraire, N. C. Nguyen, and B. Cockburn, A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations (AIAA Paper 2010-363), Proceedings of the 48th AIAA Aerospace Sciences Meeting and Exhibit (Orlando, Florida), January 2010.","DOI":"10.2514\/6.2010-363"},{"issue":"3","key":"55","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1002\/nme.2485","article-title":"A space-time discontinuous Galerkin method for the solution of the wave equation in the time domain","volume":"78","author":"Petersen, Steffen","year":"2009","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"key":"56","first-page":"292","article-title":"A mixed finite element method for 2nd order elliptic problems","author":"Raviart, P.-A.","year":"1977"},{"key":"57","unstructured":"S.-C. Soon, Hybridizable discontinuous Galerkin methods for solid mechanics, Ph.D. thesis, University of Minnesota, 2008."},{"issue":"8","key":"58","doi-asserted-by":"publisher","first-page":"1058","DOI":"10.1002\/nme.2646","article-title":"A hybridizable discontinuous Galerkin method for linear elasticity","volume":"80","author":"Soon, S.-C.","year":"2009","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"issue":"5","key":"59","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1007\/BF01397550","article-title":"A family of mixed finite elements for the elasticity problem","volume":"53","author":"Stenberg, Rolf","year":"1988","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"60","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1051\/m2an\/1991250101511","article-title":"Postprocessing schemes for some mixed finite elements","volume":"25","author":"Stenberg, Rolf","year":"1991","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"61","series-title":"Monografie Matematyczne [Mathematical Monographs], Tom 43","volume-title":"Differential inequalities","author":"Szarski, Jacek","year":"1965"},{"key":"62","doi-asserted-by":"crossref","unstructured":"Liu Y. and Wei X., Finite-difference numerical modeling with even-order accuracy in two-phase anisotropic media, Applied Geophysics 5 (2008), 107\u2013114.","DOI":"10.1007\/s11770-008-0014-6"},{"issue":"3","key":"63","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1007\/s10543-005-0021-4","article-title":"Fourth order symmetric finite difference schemes for the acoustic wave equation","volume":"45","author":"Zemui, Abraham","year":"2005","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2014-83-285\/S0025-5718-2013-02743-3\/S0025-5718-2013-02743-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-285\/S0025-5718-2013-02743-3\/S0025-5718-2013-02743-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:43:11Z","timestamp":1776793391000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-285\/S0025-5718-2013-02743-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,7,16]]},"references-count":63,"journal-issue":{"issue":"285","published-print":{"date-parts":[[2014,1]]}},"alternative-id":["S0025-5718-2013-02743-3"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2013-02743-3","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":[[2013,7,16]]}}}