In adaptive finite elements, the residuals are defined as R_k(U_T) = f + Δu_T and R_E(u_T) = -J(underlineE · ∠‡u_T) if E ∈ ε_Ω, g - underlineE ∠‡u_T if E ∈ ε_γ_N, 0 otherwise . The Poisson equation is given by Δu_T = -f and the variational form is int ∠‡u_T ∠‡v = int fv + int_⁶Ω gv for all v in H¹(Ω). After applying integration by parts and using the inequalities associated with quasi-interpolation, we have leq sum_K||R_K(u_T)||_K c_A1h_K||w||_H¹( tildew_K) + sum_E||R_E(u_T)||_Ec_A2h_E^ frac12||w||_H¹( tildew_E) (*). The next step involves the use of the Cauchy-Schwarz inequality for sums to obtain, leq textmax(c_A2, c_A1) left[ sum_Kh²_K||R_K(u_T)²_K + sum_Eh_E||R_E(u_T)||²_E right]^ frac12 cdot left[ sum_K||w||_H¹( tildew_K)² + sum_E||w||_H¹( tildew_E)² right]^ frac12 (+), where the C-S inequality is (sum_k(a_Kb_K))² leq (sum_k a²_k)(sum_k b_k²). How do we get from (*) to (+)?
