Design and Convergence of Afem in H(div)

We design an adaptive finite element method (AFEM) for mixed boundary value problems associated with the differential operator A-∇div in H(div, Ω). For A being a variable coefficient matrix with possible jump discontinuities, we provide a complete a posteriori error analysis which applies to both Raviart–Thomas ℝ𝕋n and Brezzi–Douglas–Marini 𝔹𝔻𝕄n elements of any order n in dimensions d = 2, 3. We prove a strict reduction of the total error between consecutive iterates, namely a contraction property for the sum of energy error and oscillation, the latter being solution-dependent. We present numerical experiments for ℝ𝕋n with n = 0, 1 and 𝔹𝔻𝕄1 which document the performance of AFEM and corroborate as well as extend the theory.