## 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.## References

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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