Can you show me the main differences between 2 methods? I find out 2 reasons but I don't know they are right or not.
XFEM is constructed base on enrichment functions whereas P1-bubble is constructed base on bubble functions. Basically, 2 types of these functions are different (what are these differences?).
For $P_k$-bubble: It is not necessary to consider a space of polynomials of degree of $k+1$, it just needs to add more one extra degree of freedom to the barycenter of every simplex of the triangulation $T_h$ of the domain $\Omega$. Whereas, xFEM is only considers the enrichment function at some nodes relating to the cracks (or the main problem).