Abstract We introduce a non-conforming finite element method for second order elliptic interface problems. Our approach applies to problems in which discontinuous coefficients and singular sources on the interface may give rise to jump discontinuities in either the solution or its normal derivative. Given a standard background mesh and an interface that passes between elements, the key idea is to construct a singular correction function which satisfies the prescribed jump conditions, providing accurate sub-grid resolution of the discontinuities. Utilizing the closest point extension and an implicit interface representation by the signed distance function, an algorithm is established to construct the correction function. The result is a function which is supported only on the interface elements, represented by the regular basis functions, and bounded independently of the interface location with respect to the background mesh. In the particular case of a constant second order coefficient, our regularization by singular function is straightforward, and the resulting left-hand-side is identical to that of a regular problem without introducing any instability. The influence of the regularization appears solely on the right-hand-side, which simplifies the implementation. In the more general case of discontinuous second order coefficients, a normalization is invoked which introduces a constraint equation on the interface. This results in a problem statement similar to that of a saddle-point problem. We employ two-level-iteration as the solution strategy, which exhibits aspects similar to those of iterative preconditioning strategies. Elliptic interface problems appear in many physical applications, including Stefan problems, fluids problems, materials issues, free boundary problems, and shape optimization In the above, as a model problem, we can take a Poisson equation with piecewise constant coefficient, a i > 0 on each Ω i ; −∇·(a i ∇u) = f on Ω i for i = 1, 2 with the boundary data given on ∂Ω. For well-posedness, we need additional information on the behavior of the solution on the interface. Let g and h be
