{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T05:54:38Z","timestamp":1778219678582,"version":"3.51.4"},"reference-count":32,"publisher":"American Mathematical Society (AMS)","issue":"238","license":[{"start":{"date-parts":[[2002,5,11]],"date-time":"2002-05-11T00:00:00Z","timestamp":1021075200000},"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 study the convergence properties of the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h p\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>h<\/mml:mi>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">hp<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -version of the local discontinuous Galerkin finite element method for convection-diffusion problems; we consider a model problem in a one-dimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the mesh-width\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                    , in the polynomial degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and in the regularity of the exact solution. We identify a special numerical flux for which the estimates are optimal in both\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                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The theoretical results are confirmed in a series of numerical examples.\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01317-5","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"455-478","source":"Crossref","is-referenced-by-count":195,"title":["Optimal a priori error estimates for the \u210e\ud835\udc5d-version of the local discontinuous Galerkin method for convection\u2013diffusion problems"],"prefix":"10.1090","volume":"71","author":[{"given":"Paul","family":"Castillo","sequence":"first","affiliation":[]},{"given":"Bernardo","family":"Cockburn","sequence":"additional","affiliation":[]},{"given":"Dominik","family":"Sch\u00f6tzau","sequence":"additional","affiliation":[]},{"given":"Christoph","family":"Schwab","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,5,11]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1006\/jcph.1996.5572","article-title":"A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations","volume":"131","author":"Bassi, F.","year":"1997","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"3-4","key":"2","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1016\/S0045-7825(98)00359-4","article-title":"A discontinuous \u210e\ud835\udc5d finite element method for convection-diffusion problems","volume":"175","author":"Baumann, Carlos Erik","year":"1999","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"P. Castillo, An optimal error estimate for the local discontinuous Galerkin method, Discontinuous Galerkin Methods: Theory, Computation and Applications (B. Cockburn, G.E. Karniadakis, and C.-W. Shu, eds.), Lectures Notes in Computational Science and Engineering, vol. 11, Springer-Verlag, 2000, pp. 285\u2013290.","DOI":"10.1007\/978-3-642-59721-3_23"},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/978-3-662-03882-6_2","article-title":"Discontinuous Galerkin methods for convection-dominated problems","author":"Cockburn, Bernardo","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/3540658939"},{"key":"5","doi-asserted-by":"crossref","unstructured":"B. Cockburn and C. Dawson, Some extensions of the local discontinuous Galerkin method for convection-diffusion equations in multidimensions, in Proceedings of the Conference on the Mathematics of Finite Elements and Applications: MAFELAP X (J. 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