Quasi-Optimality of Adaptive Mixed FEMs for Non-selfadjoint Indefinite Second-Order Linear Elliptic Problems

Abstract The well-posedness and the a priori and a posteriori error analysis of the lowest-order Raviart–Thomas mixed finite element method (MFEM) has been established for non-selfadjoint indefinite second-order linear elliptic problems recently in an article by Carstensen, Dond, Nataraj and Pani (Numer. Math., 2016). The associated adaptive mesh-refinement strategy faces the difficulty of the flux error control in H ⁢ ( div , Ω ) {H({\operatorname{div}},\Omega)} and so involves a data-approximation error ∥ f - Π 0 ⁢ f ∥ {\lVert f-\Pi_{0}f\rVert} in the L 2 {L^{2}} norm of the right-hand side f and its piecewise constant approximation Π 0 ⁢ f {\Pi_{0}f} . The separate marking strategy has recently been suggested with a split of a Dörfler marking for the remaining error estimator and an optimal data approximation strategy for the appropriate treatment of ∥ f - Π 0 ⁢ f ∥ L 2 ⁢ ( Ω ) {\|f-\Pi_{0}f\|_{L^{2}(\Omega)}} . The resulting strategy presented in this paper utilizes the abstract algorithm and convergence analysis of Carstensen and Rabus (SINUM, 2017) and generalizes it to general second-order elliptic linear PDEs. The argument for the treatment of the piecewise constant displacement approximation u RT {u_{{\mathrm{RT}}}} is its supercloseness to the piecewise constant approximation Π 0 ⁢ u {\Pi_{0}u} of the exact displacement u. The overall convergence analysis then indeed follows the axioms of adaptivity for separate marking. Some results on mixed and nonconforming finite element approximations on the multiply connected polygonal 2D Lipschitz domain are of general interest.