Edge stabilisation for Galerkin approximations of convection-diffusion-reaction problems

n this paper we recall a stabilization technique for finite element methods for convection-diffusion-reaction equations, originally proposed by J. Douglas, Jr. and T. Dupont [in Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975), 207--216, Lecture Notes in Phys., 58, Springer, Berlin, 1976; MR0440955 (55 \\#13823)]. The method uses least square stabilization of the gradient jumps across element boundaries. We prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates. We address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results.