Exact subgrid interface correction schemes for elliptic interface problems

We introduce a nonconforming 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 that satisfies the prescribed jump conditions, providing accurate subgrid 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 that 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 a 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.