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Stabilized Galerkin Approximation of Convection-Diffusion-Reaction Equations: Discrete Maximum Principle and Convergence
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posted on 2023-06-08, 05:08 authored by Erik Burman, Alexandre ErnWe analyze a nonlinear shock-capturing scheme for -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 -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 Pclet 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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Publication status
- Published
Journal
Mathematics of ComputationISSN
0025-5718External DOI
Issue
252Volume
74Page range
1637-1652Pages
16.0Department affiliated with
- Mathematics Publications
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- No
Peer reviewed?
- Yes
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2012-02-06Usage metrics
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