{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T21:03:30Z","timestamp":1781730210965,"version":"3.54.5"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"252","license":[{"start":{"date-parts":[[2006,6,7]],"date-time":"2006-06-07T00:00:00Z","timestamp":1149638400000},"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 analyze a nonlinear shock-capturing scheme for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -conform- ing, piecewise-affine finite element approximations of linear elliptic problems. The meshes are assumed to satisfy two standard conditions: a local quasi-uniformity property and the Xu\u2013Zikatanov condition ensuring that the stiffness matrix associated with the Poisson equation is an\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper M\">\n                        <mml:semantics>\n                          <mml:mi>M<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">M<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -matrix. A discrete maximum principle is rigorously established in any space dimension for convection-diffusion-reaction problems. We prove that the shock-capturing finite element solution converges to that without shock-capturing if the cell P\u00e9clet numbers are sufficiently small. Moreover, in the diffusion-dominated regime, the difference between the two finite element solutions super-converges with respect to the actual approximation error. Numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates.\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01761-8","type":"journal-article","created":{"date-parts":[[2005,8,10]],"date-time":"2005-08-10T10:23:21Z","timestamp":1123669401000},"page":"1637-1652","source":"Crossref","is-referenced-by-count":99,"title":["Stabilized Galerkin approximation of convection-diffusion-reaction equations: discrete maximum principle and convergence"],"prefix":"10.1090","volume":"74","author":[{"given":"Erik","family":"Burman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alexandre","family":"Ern","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2005,6,7]]},"reference":[{"issue":"219","key":"1","doi-asserted-by":"publisher","first-page":"935","DOI":"10.1090\/S0025-5718-97-00848-X","article-title":"A least-squares approach based on a discrete minus one inner product for first order systems","volume":"66","author":"Bramble, James H.","year":"1997","journal-title":"Math. 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