{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T01:50:18Z","timestamp":1774921818650,"version":"3.50.1"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"242","license":[{"start":{"date-parts":[[2003,11,20]],"date-time":"2003-11-20T00:00:00Z","timestamp":1069286400000},"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>We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Delta x\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mi mathvariant=\"normal\">\u0394<\/mml:mi>\n      <mml:mi>x<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\Delta x<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> only. For example, when polynomials of degree <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> are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k plus 1 slash 2\">\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 class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mn>2<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">k+1\/2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> in the <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>-norm, whereas the post-processed approximation is of order <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>; if the exact solution is in <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> only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k plus 1 slash 2\">\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 class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mn>2<\/mml:mn>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">k+1\/2<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> in <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared left-parenthesis normal upper Omega 0 right-parenthesis\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:msup>\n        <mml:mi>L<\/mml:mi>\n        <mml:mn>2<\/mml:mn>\n      <\/mml:msup>\n      <mml:mo stretchy=\"false\">(<\/mml:mo>\n      <mml:msub>\n        <mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi>\n        <mml:mn>0<\/mml:mn>\n      <\/mml:msub>\n      <mml:mo stretchy=\"false\">)<\/mml:mo>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">L^2(\\Omega _0)<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>, where <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Omega 0\">\n  <mml:semantics>\n    <mml:msub>\n      <mml:mi mathvariant=\"normal\">\u03a9<\/mml:mi>\n      <mml:mn>0<\/mml:mn>\n    <\/mml:msub>\n    <mml:annotation encoding=\"application\/x-tex\">\\Omega _0<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.<\/p>","DOI":"10.1090\/s0025-5718-02-01464-3","type":"journal-article","created":{"date-parts":[[2003,2,10]],"date-time":"2003-02-10T16:10:52Z","timestamp":1044893452000},"page":"577-606","source":"Crossref","is-referenced-by-count":135,"title":["Enhanced accuracy by post-processing for finite element methods for hyperbolic equations"],"prefix":"10.1090","volume":"72","author":[{"given":"Bernardo","family":"Cockburn","sequence":"first","affiliation":[]},{"given":"Mitchell","family":"Luskin","sequence":"additional","affiliation":[]},{"given":"Chi-Wang","family":"Shu","sequence":"additional","affiliation":[]},{"given":"Endre","family":"S\u00fcli","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2002,11,20]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1090\/psapm\/056\/1718897","article-title":"Computational methods for singularly perturbed systems","author":"Adjerid, Slimane","year":"1999"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"520","DOI":"10.1137\/S0036139993269345","article-title":"High-order finite element methods for singularly perturbed elliptic and parabolic problems","volume":"55","author":"Adjerid, Slimane","year":"1995","journal-title":"SIAM J. 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