Projected gradient stabilization of sharp and diffuse interface formulations in unfitted Nitsche finite element methods

We introduce an unfitted Nitsche finite element method with a new ghost-penalty stabilization based on local projection of the solution gradient. The proposed ghost-penalty operator is straightforward to implement, ensures algebraic stability, provides an implicit extension of the solution beyond the physical domain, and stabilizes the numerical method for problems dominated by transport phenomena. This paper presents both a sharp interface version of the method and an alternative diffuse interface formulation designed to avoid integration over implicitly defined embedded surfaces. A complete numerical analysis of the sharp interface version is provided. The results of several numerical experiments support the theoretical analysis and illustrate the performance of both variants of the method.