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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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Stabilized Galerkin approximation of convection-diffusion-reaction equations: discrete maximum principle and convergence
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by Erik Burman and Alexandre Ern PDF
Math. Comp. 74 (2005), 1637-1652 Request permission

Abstract:

We analyze a nonlinear shock-capturing scheme for $H^1$-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–Zikatanov condition ensuring that the stiffness matrix associated with the Poisson equation is an $M$-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 Péclet 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.
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Additional Information
  • Erik Burman
  • Affiliation: Ecole Polytechnique Federale de Lausanne, Institute of Analysis and Scientific Computing, 1015 Lausanne, Switzerland
  • MR Author ID: 602430
  • Email: Erik.Burman@epfl.ch
  • Alexandre Ern
  • Affiliation: CERMICS, Ecole nationale des ponts et chaussées, 6 et 8, avenue B. Pascal, 77455 Marne la Vallée cedex 2, France
  • MR Author ID: 349433
  • Email: ern@cermics.enpc.fr
  • Received by editor(s): February 18, 2003
  • Received by editor(s) in revised form: August 16, 2004
  • Published electronically: June 7, 2005
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 74 (2005), 1637-1652
  • MSC (2000): Primary 65N12, 65N30, 76R99
  • DOI: https://doi.org/10.1090/S0025-5718-05-01761-8
  • MathSciNet review: 2164090