The authors present a new approach to treat numerically elliptic partial differential equations (PDEs) with discontinuous diffusion coefficients. The standard approach to numerically treating such problems using finite element methods is to assume that the discontinuities lie on the boundaries of the cells in the initial triangulation. However, this does not match applications where discontinuities occur on curves, surfaces, or manifolds, and could even be unknown beforehand. One of the obstacles to treating such discontinuity problems is that the usual perturbation theory for elliptic PDEs assumes bounds for the distortion of the coefficients in the \(L_\infty\) norm and this in turn requires that the discontinuities are matched exactly when the coefficients are approximated. The approach of the authors is based on distortion of the coefficients in an \(L_q\) norm with \(q<\infty\) which therefore does not require the exact matching of the discontinuities. This new distortion theory is used to formulate new adaptive finite element methods (AFEMs) for such discontinuity problems. Such AFEMs are shown to be optimal in the sense of distortion versus number of computations.