Abstarct: Elliptic partial diferential equations (PDEs) with discontinuous difusion coeficients occur in application domains such as difusions through porous media, electromagnetic field propagation on heterogeneous media, and difusion processes on rough surfaces. 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 coeficients in the L1 norm and this in turn requires that the discontinuities are matched exactly when the coeficients are approximated. We present a new approach baxsed on distortion of the coeficients in an Lq norm with q < 1 which therefore does not require the exact matching of the discontinuities. We then use this new distortion theory to formulate new adaptive finite element methods (AFEMs) for such discontinuity problems. We show that such AFEMs are optimal in the sense of distortion versus number of computations, and report insightful numerical results supporting our analysis.