  The state-of-the art proof of a global inf-sup condition on mixed finite
element schemes does not allow for an analysis of truly indefinite,
second-order linear elliptic PDEs. This paper, therefore, first analyses a
nonconforming finite element discretization which converges owing to some a
priori $L^2$ error estimates even for reduced regularity on non-convex
polygonal domains. An equivalence result of that nonconforming finite element
scheme to the mixed finite element method (MFEM) leads to the well-posedness of
the discrete solution and to a priori error estimates for the MFEM. The
explicit residual-based a posteriori error analysis allows some reliable and
efficient error control and motivates some adaptive discretization which
improves the empirical convergence rates in three computational benchmarks.
