This thesis covers two aspects of first-order system least-squares methods (FOSLS). In the first part, we study methods of adaptive mesh refinement (AMR) that are applicable to FOSLS. In the second part, which comprises the larger part of this thesis, we devise a FOSLS $L^2$ formulation for the solution of diffusion equations with discontinuous coefficients. We develop a finite element discretization that incorporates singular basis functions that model singular behavior of solutions accurately, derive error estimates, and present a two-grid solver for the discrete problem.