@article{discovery1384765,
         journal = {Mathematics of Computation},
            note = {Copyright {\copyright} 2005 American Mathematical Society. The copyright for this article reverts to public domain 28 years after publication.},
           title = {Stabilized Galerkin approximation of convection-diffusion-reaction equations: Discrete maximum principle and convergence},
           pages = {1637--1652},
          volume = {74},
            year = {2005},
          number = {252},
           month = {June},
             url = {http://dx.doi.org/10.1090/S0025-5718-05-01761-8},
            issn = {0025-5718},
        abstract = {We analyze a nonlinear shock-capturing scheme for H1-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 P{\'e}clet 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.},
          author = {Burman, E and Ern, A}
}