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Mathematics of Computation

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Stabilized Galerkin approximation of convection-diffusion-reaction equations: discrete maximum principle and convergence

Authors: Erik Burman and Alexandre Ern
Journal: Math. Comp. 74 (2005), 1637-1652
MSC (2000): Primary 65N12, 65N30, 76R99
Published electronically: June 7, 2005
MathSciNet review: 2164090
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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

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

Received by editor(s): February 18, 2003
Received by editor(s) in revised form: August 16, 2004
Published electronically: June 7, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.