Tamtam Proceedings - lamsin

178 Achchab et al.1. IntroductionDuring the last two decades there has been a rapid development in practical a posteriorierror estimation techniques for elliptic PDE’s. This explosion of interest has been drivenby the underlying need to increase the reliability and efficiency of finite element softwarefor solving such problems. In the specific case of the convection-diffusion equations, thework of Verfürth [9] is considered as the state of the art for residual based a posteriori errorestimates for reaction-convection-diffusion equations. The stabilazation was introducedby SUPG [5],[8], in isotropic finite element case. For anisotropic case [3], Achchaband al [1] derive lower and upper bounds. The quotient of lower and upper bounds isproportional to the matching function which depends on the anisotropy of the problemand is independent of the small perturbation parameter ɛ.For more details see Kunert [7] .We consider the reaction-convection-diffusion equation−ε∆u + β∇u + σu = f dans Ωu = 0 sur Γ D (1)∂ n u = g sur Γ N .Where Ω ⊂ IR n , is a bounded domain with the boundary ∂Ω = Γ D ∪ Γ N such thatΓ D ∩ Γ N = ∅. We suppose the following assumptions (H)• 0 < ε