Tamtam Proceedings - lamsin

180 Achchab et al.such that}Xh 2 ={v h 2 ∈ HD(Ω) 1 : vh|T 2 ∈ 〈ψ T 〉 , ∀ T ∈ T h(7)where ψ T = ̂ψ(G T ) and G T : T −→ ̂T is one to one affine mapping that maps T into ̂Tand ̂ψ is the bubble function defined on the reference simplex ̂T .We define c ψ =∫̂T ̂ψ 2meas( ̂T ), one easily verifies that{ ∫T ψ2 = c ψ meas(T ),c ψ meas(T ) ≤ ∫ T ψ ≤ meas(T )Finally the discrete problems we consider is to find u h ∈ X h so thata(u h , v h ) + b h (u 2 h, v 2 h) = (f, v h ) + (g, v h ) ΓN ∀v h ∈ X h (8)This problem has a unique solution [6].3. A posteriori error estimationLet f h and g h be approximations of f and g respectively by P 1 piecewise polynomialswith respect to T h and with respect to Γ N induced by T h . SetWhereη 2 T,h = α 2 T ‖R T,h (u h )‖ 2 0,T+ αT 2 ‖∇u 2 h‖ 2 0,T+ 1 ∑2+E∈E T ∩E h,Ωε −1∑E∈E T ∩E h,Nε −12 αE ‖R E,h (u h )‖ 2 0,E2 αE ‖R E,h (u h )‖ 2 0,E (9)R T,h (u h ) = f h + ε△u h − β∇u h − σu h . (10)⎧⎨ −[ε∂ n u h ] E si E ∈ E h,ΩR E,h (u h ) = g h − ε∂ n u h si E ∈ E h,N(11)⎩0 si E ∈ E h,DThus we have established the following result.TAMTAM –Tunis– 2005