| We present a non-conforming finite element method for second order elliptic interface problems with discontinuous coefficients and singular sources, which are equivalently described by the jump discontinuities in the solution and the normal derivatives. In the first step, the second order coefficient of a given problem is normalized resulting in the Laplacian and a constraint introduced on the interface. In the second step, the normalized problem is regularized by a singular correction function which satisfies the prescribed jump conditions. This allows us to apply a regular finite element method on meshes non-conforming to the interface. The combined normalization-regularization exhibits aspects similar to those of iterative preconditioning strategies. Utilizing the closest point extension and the interface representation by the signed distance function, an algorithm is established to construct the correction function. The result is a function 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. |
