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

Burman, Erik; Ern, Alexandre

doi:10.1090/S0025-5718-05-01761-8

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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