Stabilized Galerkin Approximation of Convection-Diffusion-Reaction Equations: Discrete Maximum Principle and Convergence

Burman, Erik and Ern, Alexandre (2005) Stabilized Galerkin Approximation of Convection-Diffusion-Reaction Equations: Discrete Maximum Principle and Convergence. Mathematics of Computation, 74 (252). pp. 1637-1652. ISSN 0025-5718

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Abstract

We 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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Depositing User: Erik Burman
Date Deposited: 06 Feb 2012 20:08
Last Modified: 04 Apr 2012 11:59
URI: http://sro.sussex.ac.uk/id/eprint/24263
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