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Elliptic partial differential equations (PDEs) with discontinuous diffusioncoefficients occur in application domains such as diffusions through porousmedia, electro-magnetic field propagation on heterogeneous media, and diffusionprocesses on rough surfaces. The standard approach to numerically treating suchproblems using finite element methods is to assume that the discontinuities lieon the boundaries of the cells in the initial triangulation. However, this doesnot match applications where discontinuities occur on curves, surfaces, ormanifolds, and could even be unknown beforehand. One of the obstacles totreating such discontinuity problems is that the usual perturbation theory forelliptic PDEs assumes bounds for the distortion of the coefficients in the$L_\infty$ norm and this in turn requires that the discontinuities are matchedexactly when the coefficients are approximated. We present a new approach basedon distortion of the coefficients in an $L_q$ norm with $q<\infty$ whichtherefore does not require the exact matching of the discontinuities. We thenuse this new distortion theory to formulate new adaptive finite element methods(AFEMs) for such discontinuity problems. We show that such AFEMs are optimal inthe sense of distortion versus number of computations, and report insightfulnumerical results supporting our analysis.

Adaptive Finite Element Methods for Elliptic Problems with Discontinuous Coefficients