This paper studies adaptive finite element methods (AFEMs), based on piecewise linear elements and newest vertex bisection, for solving second order elliptic equations with piecewise constant coefficients on a polygonal domain 2 . The main contribution is to build algorithms that hold for a general right-hand side fH -1 (). Prior work assumes almost exclusively that fL 2 (). New data indicators based on local H -1 norms are introduced, and then the AFEMs are based on a standard bulk chasing strategy (or Drfler marking) combined with a procedure that adapts the mesh to reduce these new indicators. An analysis of our AFEM is given which establishes a contraction property and optimal convergence rates N -s with 0
