{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:38:05Z","timestamp":1776728285795,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"238","license":[{"start":{"date-parts":[[2002,5,22]],"date-time":"2002-05-22T00:00:00Z","timestamp":1022025600000},"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 consider a family of tensor product finite element methods for hyperbolic equations in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper R Superscript upper N\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>N<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">R^{N}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N greater-than-or-equal-to 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">N\\ge 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , which are explicit and generate a continuous approximate solution. The base case\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N equals 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">N=2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (an extension of the box scheme to higher order) is due to Winther, who proved stability and optimal order convergence. By means of a simple counterexample, we show that, for linear approximation with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N greater-than-or-equal-to 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">N \\ge 3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the corresponding methods are unstable.\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01334-5","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"527-535","source":"Crossref","is-referenced-by-count":0,"title":["On the stability of a family of finite element methods for hyperbolic problems"],"prefix":"10.1090","volume":"71","author":[{"given":"Gerard","family":"Richter","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,5,22]]},"reference":[{"issue":"3","key":"1","doi-asserted-by":"publisher","first-page":"1127","DOI":"10.1137\/S0036142994264882","article-title":"Analysis of the cell-vertex finite volume method for hyperbolic problems with variable coefficients","volume":"34","author":"Balland, Philippe","year":"1997","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5","key":"2","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1002\/fld.1650140502","article-title":"Cell vertex finite volume discretizations in three dimensions","volume":"14","author":"Crumpton, P. I.","year":"1992","journal-title":"Internat. J. Numer. Methods Fluids","ISSN":"https:\/\/id.crossref.org\/issn\/0271-2091","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1137\/0724021","article-title":"Analysis of a continuous finite element method for hyperbolic equations","volume":"24","author":"Falk, Richard S.","year":"1987","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"730","DOI":"10.1137\/0729046","article-title":"Local error estimates for a finite element method for hyperbolic and convection-diffusion equations","volume":"29","author":"Falk, Richard S.","year":"1992","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"5","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"},{"issue":"173","key":"6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2008211","article-title":"An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation","volume":"46","author":"Johnson, C.","year":"1986","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"7","first-page":"89","article-title":"On a finite element method for solving the neutron transport equation","author":"Lasaint, P.","year":"1974"},{"issue":"220","key":"8","doi-asserted-by":"publisher","first-page":"1389","DOI":"10.1090\/S0025-5718-97-00886-7","article-title":"Analysis of a cell-vertex finite volume method for convection-diffusion problems","volume":"66","author":"Morton, K. W.","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"9","unstructured":"W. H. Reed and T. R. Hill, Triangular mesh methods for solving the neutron transport equation, Los Alamos Scientific Laboratory Report LA-UR-73-479.."},{"key":"10","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1137\/0108040","article-title":"On centered difference equations for hyperbolic systems","volume":"8","author":"Wendroff, Burton","year":"1960","journal-title":"J. Soc. Indust. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0368-4245","issn-type":"print"},{"issue":"153","key":"11","doi-asserted-by":"publisher","first-page":"65","DOI":"10.2307\/2007726","article-title":"A stable finite element method for initial-boundary value problems for first-order hyperbolic systems","volume":"36","author":"Winther, Ragnar","year":"1981","journal-title":"Math. 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