{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T19:09:09Z","timestamp":1782932949977,"version":"3.54.5"},"reference-count":46,"publisher":"American Mathematical Society (AMS)","issue":"301","license":[{"start":{"date-parts":[[2017,1,11]],"date-time":"2017-01-11T00:00:00Z","timestamp":1484092800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We analyze the discontinuous Galerkin method in time combined with a finite element method with symmetric stabilization in space to approximate evolution problems with a linear, first-order differential operator. A unified analysis is presented for space discretization, including the discontinuous Galerkin method and\n \n \n \n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n H^1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -conforming finite elements with interior penalty on gradient jumps. Our main results are error estimates in various norms for smooth solutions. Two key ingredients are the post-processing of the fully discrete solution by lifting its jumps in time and a new time-interpolate of the exact solution. We first analyze the\n \n \n \n \n \n L<\/mml:mi>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:msup>\n (<\/mml:mo>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n L^\\infty (L^2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n (at discrete time nodes) and\n \n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n (<\/mml:mo>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n L^2(L^2)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n errors and derive a superconvergent bound of order\n \n \n \n \n (<\/mml:mo>\n \n \n \u03c4\n \n <\/mml:mi>\n \n k<\/mml:mi>\n +<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n \n h<\/mml:mi>\n \n r<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n (\\tau ^{k+2}+h^{r+1\/2})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for static meshes for\n \n \n \n \n k<\/mml:mi>\n \n \u2265\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n k\\ge 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Here,\n \n \n \n \n \u03c4\n \n <\/mml:mi>\n \\tau<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the time step,\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the polynomial order in time,\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the size of the space mesh, and\n \n \n \n r<\/mml:mi>\n r<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n the polynomial order in space. For the case of dynamically changing meshes, we derive a novel bound on the resulting projection error. Finally, we prove new optimal bounds on static meshes for the error in the time-derivative and in the discrete graph norm.\n <\/p>","DOI":"10.1090\/mcom\/3073","type":"journal-article","created":{"date-parts":[[2015,10,21]],"date-time":"2015-10-21T10:08:33Z","timestamp":1445422113000},"page":"2099-2129","source":"Crossref","is-referenced-by-count":17,"title":["Discontinuous Galerkin method in time combined with a stabilized finite element method in space for linear first-order PDEs"],"prefix":"10.1090","volume":"85","author":[{"given":"Alexandre","family":"Ern","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Friedhelm","family":"Schieweck","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2016,1,11]]},"reference":[{"issue":"21-22","key":"1","doi-asserted-by":"publisher","first-page":"1747","DOI":"10.1016\/j.cma.2011.02.003","article-title":"Discontinuous Galerkin time stepping with local projection stabilization for transient convection-diffusion-reaction problems","volume":"200","author":"Ahmed, N.","year":"2011","journal-title":"Comput. 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