%V 74
%P 1637-1652
%O Copyright © 2005 American Mathematical Society. The copyright for this article reverts to public domain 28 years after publication.
%D 2005
%X 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 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.
%A E Burman
%A A Ern
%T Stabilized Galerkin approximation of convection-diffusion-reaction equations: Discrete maximum principle and convergence
%J Mathematics of Computation
%N 252
%L discovery1384765