## 6 Numerical study on the error and applications to planar-symmetric problems
### On the interpolation in the blending elements  
The treatment of blending elements (i.e. those whose nodal support is only partially enriched, see Section 4.1) is discussed here by means of a numerical example. Different approaches to the temperature interpolation in this region are possible: (i) the partial enrichment by routinely applying Eq. (31), (ii) the adoption of special strategies such as the use of hierarchical functions (e.g. [62]), and (iii) the use of standard FE shape functions only (as in the non-enriched elements). Figure 12: Temperature field in the vicinity of the interface. The XFEM strategy consists in enriching the nodes whose support is intersected by the discontinuity. In this way the subset \(A\) of the standard FE shape functions constitutes a local partition of unity in the region \(\Omega_{{}_{\Gamma}}\) surrounding the discontinuity and does not affect the region far from the enriched nodes (where standard FE interpolation holds). However, by following approach (i), the enrichment involves a region \(\Omega_{{}_{A}}\!\supseteq\!\Omega_{{}_{\Gamma}}\) and is only partial in the region \(\Omega_{{}_{A}}\!\setminus\!\Omega_{{}_{\Gamma}}\), i.e. in the blending elements. In a blending element, only some nodes are enriched and the subset of the standard shape functions used to weight the discontinuity does not represent a partition of unity. This leads to the appearance of pathological terms and, in general, to a sub-optimal convergence rate [62]. One possible cure, the approach (ii), is the use of hierarchical, higher-order shape functions as proposed in Reference [62]. The idea is that the enrichment functions are obtained by the product of the standard FE shape functions (linear, in our case) and the enrichment functions (also linear); therefore, it is possible to compensate their pathological contributions by using higher order FE shape functions in the blending elements (second grade polynomials). This approach proves very effective as demonstrated by their proposer. However, in those cases in which the interface is not fixed, local modifications of the topology are required as the interface evolves. When the interface moves from one element to another one, new nodes must be enriched and, consequently, some standard elements become blending elements. There, in a topological approach, new hierarchical nodes must be added. This may complicate the implementation of the method, even though the result is computationally cheaper than the remeshing required by standard FEM with adaptive mesh. [PERSON] et al. [62], shows that the approximation error \(\varepsilon\) in a two-node one-dimensional blending element satisfies the inequality  
\[\max\left|\varepsilon\left(y\right)\right|\!\leq\!\frac{1}{8}\ell_{\mathrm{e}} ^{2}\max\left|\left[\frac{d^{2}T}{dx^{2}}\right]_{x=y}+\frac{2\alpha}{\ell_{ \mathrm{e}}}\frac{d\zeta}{dx}\left(\overline{x}\right)\right|, \tag{60}\]  
where \(y\) and \(\overline{x}\) are points inside the element and \(a\) is the extended degree of freedom of the enriched node. It is clear that the second term in the right-hand side of Eq. (60), which is not present in the standard FE case, may lead to an increased error. This pathological term can be compensated by using higher-order (second grade, in our case) standard shape functions for the blending element, controlled by additional standard degrees of freedom (hierarchical approach). Alternatively, as in the approach (iii), it is possible to eliminate the pathological term by considering the enrichment functions only in those elements whose support is completely enriched and simply using the standard FE shape functions in the blending elements. In fracture mechanics, where the XFEM has been originally developed, this approach causes problems, as the displacement field may not be well reproduced in the vicinity of the interface boundaries (the crack tips) and conformity between enriched and blending elements may be lost. In the case of Stephan problem, however, these are not concerns.