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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Discontinuous Galerkin method in time combined with a stabilized finite element method in space for linear first-order PDEs
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by Alexandre Ern and Friedhelm Schieweck PDF
Math. Comp. 85 (2016), 2099-2129 Request permission

Abstract:

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 $H^1$-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 $L^\infty (L^2)$ (at discrete time nodes) and $L^2(L^2)$ errors and derive a superconvergent bound of order $(\tau ^{k+2}+h^{r+1/2})$ for static meshes for $k\ge 1$. Here, $\tau$ is the time step, $k$ the polynomial order in time, $h$ the size of the space mesh, and $r$ 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.
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Additional Information
  • Alexandre Ern
  • Affiliation: Universite Paris-Est, CERMICS (ENPC), 77455 Marne la Vallee Cedex 2, France
  • MR Author ID: 349433
  • Email: ern@cermics.enpc.fr
  • Friedhelm Schieweck
  • Affiliation: Institut für Analysis und Numerik, Otto–von–Guericke–Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany
  • MR Author ID: 155960
  • Email: schiewec@ovgu.de
  • Received by editor(s): February 18, 2014
  • Received by editor(s) in revised form: April 14, 2015
  • Published electronically: January 11, 2016
  • © Copyright 2016 American Mathematical Society
  • Journal: Math. Comp. 85 (2016), 2099-2129
  • MSC (2010): Primary 65M12, 65M60; Secondary 65J10
  • DOI: https://doi.org/10.1090/mcom/3073
  • MathSciNet review: 3511276