Adaptive mixed finite element methods for non-self-adjoint indefinite second-order elliptic pdes with optimal rates

This paper establishes the convergence of adaptive mixed finite element methods for second-order linear non-self-adjoint indefinite elliptic problems in three dimensions with piecewise Lipschitz continuous coefficients. The error is measured in the L² norms and then allows for an adaptive algorithm with collective Dörfler marking. The axioms of adaptivity apply to this setting and guarantee the rate optimality for Raviart–Thomas and Brezzi–Douglas–Marini finite elements of any order for sufficiently small initial mesh-sizes and bulk parameter. The proofs require some L² best-approximation property from the medius analysis of mixed finite element methods and several supercloseness results.