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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.

  1. 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?).

  2. 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).

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The difference between bubble-enriched and XFEM-enriched spaces is that in the former case, you do the same thing on the same cell (i.e., you enrich with shape functions created with the same construction on every cell), whereas in the latter you add shape functions that are defined globally, independent of the mesh. The former therefore fits well into the usual finite element implementations: they work well as they do the same thing on every cell. On the other hand, the XFEM is difficult to implement because it requires doing something different on every cell.

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