Enriched and unfitted finite element methods are powerful extensions of the popular classical Finite Element Method (FEM). In FEM, piecewise polynomial functions defined over geometrically simple elements are used to represent both the geometry of the computational domain and the solution of the boundary value problem of interest. Over the years, researchers have identified limitations to this framework.

(i) Meshing complex domains may be difficult, numerically costly and may lack robustness; 

(ii) Adaptive modeling requires numerous re-meshing operations during the solution process in order to improve the accuracy and/or stability of the numerical simulation. These re-meshing cycles can be computationally expensive and should be avoided; 

(iii) FE solvers for boundary value problems whose solutions exhibit singularities or discontinuities are slow to converge with mesh refinement. This is particularly true for higher-order methods, where sharp solution features may trigger instabilities