Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes

We derive a nonlinear stabilized Galerkin approximation of the Laplace operator for which we prove a discrete maximum principle on arbitrary meshes and for arbitrary space dimension without resorting to the well-known acute condition or generalizations thereof. We also prove the existence of a discrete solution and discuss the extension of the scheme to convection-diffusion-reaction equations. Finally, we present examples showing that the new scheme cures local minima produced by the standard Galerkin approach while maintaining first-order accuracy in the H-1-norm. (C) 2004 Academie des sciences. Published by Elsevier SAS. All rights reserved.