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 H1-norm.