In this section the numerical behavior of the different approximation spaces introduced previously is investigated through a couple of benchmarks. First of all, we assess the asymptotic behavior of strongly discontinuous approximations, through a one dimensional case. Secondly, we consider the convergence study of a twodimensional problem with a crack and singular approximation.
Numerical behavior of strongly discontinuous approximations
Numerical analysis of strongly discontinuous approximations with linear elements
Let us consider the basic case of the traction of a onedimensional bar with a fictitious bimaterial interface. The arbitrary interface is positioned along the abscissa \(x\left( \varepsilon \right) \) (Fig. 8). The problem even though continuous is treated as a discontinuous one, with gluing interface boundary conditions imposed through a Lagrange multiplier. Actually, if the materials were different on each side of the interface, the resulting problem would be equivalent to enforcing a weak discontinuity with a strong discontinuity framework. This artefact is used so as to establish convergence results in energy norm. If it was not the case, the solution obtained would be that of two rigid bodies with prescribed displacements on one of their end.

The problem is the solution of the following differential equation:
$$\begin{aligned} {u''(x)=0}\quad \forall x\in \Omega =\left[ {0,3} \right] \end{aligned}$$
(30)

With arbitrary Dirichlet boundary conditions:
$$\begin{aligned} {u(0)=\beta }\quad {u(3)=3\alpha +\beta } \end{aligned}$$
(31)

And with continuity conditions at the interface expressed in terms of the continuity of the derivative of the displacement field (stress continuity at the interface):
$$\begin{aligned} \left. {\frac{du}{dx}} \right _{x\left( {\varepsilon ^{+}} \right) } =\alpha ^{}\quad \quad \left. {\frac{du}{dx}} \right _{x\left( {\varepsilon ^{}} \right) } =\alpha ^{+}\quad \quad \alpha ^{}=\alpha ^{+}=\alpha \end{aligned}$$
(32)
This continuity condition results from the equivalent Lagrangian form of (30) in which a Lagrange multiplier \(\lambda \) is used to impose a continuous displacement across the interface. \(\left( {u,\lambda } \right) \) is the saddle point of the following functional:
$$\begin{aligned} L\left( {u,\lambda } \right) =\frac{1}{2}\int \limits _0^3 {\left( {u^{\prime }} \right) ^{2}dx+\lambda \left[ {u\left( {x\left( {\varepsilon ^{+}} \right) } \right) u\left( {x\left( {\varepsilon ^{}} \right) } \right) } \right] } =0 \end{aligned}$$
The expected continuous solution satisfying the boundary conditions above, is:
$$\begin{aligned} u(x)=\alpha x+\beta \;,\;\forall x\in \left[ {0,3} \right] \end{aligned}$$
(33)
The resulting weak form of the problem is:
$$\begin{aligned}&\hbox {Find},\;\;\;\left. {{\begin{array}{l} {{u\in \prod }:\left\{ {w\in H^{1}/w(0)=\beta ,w(3)=3\alpha +\beta } \right\} ,\lambda \in \mathfrak {R}} \\ {{\forall v\in \prod }_0 :\left\{ {w\in H^{1}/w(0)=0,w(3)=0} \right\} } \\ {{\forall \mu \in {\mathbb {R}} }} \\ \end{array} }} \right\} \nonumber \\&\quad \mapsto \left\{ {{\begin{array}{l} {\int _0^3 {u'v'dx} =0} \\ {\left( {\lambda \alpha ^{+}} \right) v\left( {x\left( {\varepsilon ^{+}} \right) } \right) \left( {\lambda \alpha ^{}} \right) v\left( {x\left( {\varepsilon ^{}} \right) } \right) =0} \\ {\mu \left[ {u\left( {x\left( {\varepsilon ^{+}} \right) } \right) u\left( {x\left( {\varepsilon ^{}} \right) } \right) } \right] =0} \\ \end{array} }} \right. \end{aligned}$$
(34)
The XFEM discrete linear space is:
$$\begin{aligned} {\prod }_h :\left\{ {{\begin{array}{l} {w/w=a_1 \Phi _1 +a_2 \Phi _2 +b_2 H\Phi _2 +a_3 \Phi _3 +b_3 H\Phi _3 +a_4 \Phi _4 } \\ {a_1 =\beta \quad a_4 =3\alpha +\beta } \\ \end{array} }\;} \right\} \end{aligned}$$
(35)
Then, the \(6\times 6\) matrix associated with the discretization of the weak form on the XFEM space of linear functions (first column of Table 1) is:
$$\begin{aligned} \left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1&{} {1}&{} 1&{} 0&{} 0&{} 0 \\ {1}&{} 2&{} {2+2\varepsilon }&{} {1}&{} {12\varepsilon }&{} 0 \\ 1&{} {2+2\varepsilon }&{} 2&{} {12\varepsilon }&{} {1}&{} 0 \\ 0&{} {1}&{} {12\varepsilon }&{} 2&{} {2\varepsilon }&{} {1} \\ 0&{} {12\varepsilon }&{} {1}&{} {2\varepsilon }&{} 2&{} {1} \\ 0&{} 0&{} 0&{} {1}&{} {1}&{} 1 \\ \end{array} }} \right] \left\{ {{\begin{array}{l} {a_1 } \\ {a_2 } \\ {b_2 } \\ {a_3 } \\ {b_3 } \\ {a_4 } \\ \end{array} }} \right\} \end{aligned}$$
(36)
With the domain formulation of the second column of Table 1, the discrete matrix is:
$$\begin{aligned} \left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1&{} {1}&{} 0&{} 0&{} 0&{} 0 \\ {1}&{} {2\varepsilon }&{} 0&{} {1+\varepsilon }&{} 0&{} 0 \\ 0&{} 0&{} \varepsilon &{} 0&{} {\varepsilon }&{} 0 \\ 0&{} {1+\varepsilon }&{} 0&{} {1\varepsilon }&{} 0&{} 0 \\ 0&{} 0&{} {\varepsilon }&{} 0&{} {1+\varepsilon }&{} {1} \\ 0&{} 0&{} 0&{} 0&{} {1}&{} 1 \\ \end{array} }} \right] \left\{ {{\begin{array}{l} {\alpha _1 } \\ {\alpha _{2,1} } \\ {\alpha _{2,2} } \\ {\alpha _{3,1} } \\ {\alpha _{3,2} } \\ {\alpha _4 } \\ \end{array} }} \right\} \end{aligned}$$
(37)
With the shifted formulation of the third column of Table 1, the discrete matrix is:
$$\begin{aligned} \left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 1&{} {1}&{} 0&{} 0&{} 0&{} 0 \\ {1}&{} 2&{} {2\varepsilon }&{} {1}&{} {22\varepsilon }&{} 0 \\ 0&{} {2\varepsilon }&{} {4\varepsilon }&{} {2\varepsilon }&{} 0&{} 0 \\ 0&{} {1}&{} {2\varepsilon }&{} 2&{} {2+2\varepsilon }&{} {1} \\ 0&{} {22\varepsilon }&{} 0&{} {2+2\varepsilon }&{} {44\varepsilon }&{} 0 \\ 0&{} 0&{} 0&{} {1}&{} 0&{} 1 \\ \end{array} }} \right] \left\{ {{\begin{array}{l} {c_1 } \\ {c_2 } \\ {d_2 } \\ {c_3 } \\ {d_3 } \\ {c_4 } \\ \end{array} }} \right\} \end{aligned}$$
(38)
We enforce boundary conditions directly on the discrete problem, as equality constraints through Lagrange multipliers. The position of the interface depends only on \(\varepsilon \). Nonetheless, the solution does not depend on the \(\varepsilon \) parameter, so that the relative error does not depend on \(\varepsilon \), which makes our analysis relevant.
Let us consider the following arbitrary numerical values: \({\alpha ={10}/9}\) and \({\beta } =10.\)
Then, the solution is computed in standard 64bit arithmetic, with the linear solver UMFPACK, to reproduce the behavior of direct solvers.
The numerical error has to be close to zero, for any given position of the interface. The error is evaluated here in terms of the H\(^{1}\)norm:
$$\begin{aligned} \left\ {uu_h } \right\ _{H^{1}} =\sqrt{\int _0^3 {\left[ {\left( {uu_h } \right) ^{2}+\left( {{u}'{u}'_h } \right) ^{2}} \right] dx} } \end{aligned}$$
(39)
Although all three formulations represent the same approximation space, they do not have exactly the same numerical behavior (Fig. 9), at least concerning the error in H\(^{1}\)norm. The XFEM error increases steadily while the “shifted” formulation and the “domain” formulation levels of error stay close to machine precision (about \(2.2 \cdot 10^{16}\)).
Moreover, all three enrichment conditionings increase as the distance between the interface and node \(\hbox {N}_{3}\) decreases (Fig. 10). This means that all three enrichment strategies are very sensitive to the position of the interface. A preconditioner is needed for the three enrichments. In the case of XFEM, we studied Béchet et al. [21] preconditioner. For other formulations we used a diagonal preconditioner to scale the d.o.f.
The diagonal preconditioner allows a scaling of rows and columns through a multiplication with a diagonal matrix to the left and to the right:
$$\begin{aligned} Ku=f\rightarrow K'u'=f'\;\hbox {with}\;\left\{ {\begin{array}{l} K'=D_c KD_c \\ u=D_c u' \\ f'=D_c f \\ \end{array}} \right. \end{aligned}$$
(40)
In linear elasticity, \(D_c \) is usually defined as,
$$\begin{aligned} \left[ {D_c } \right] _{i,i} =\frac{1}{\sqrt{K_{i,i} }}\sqrt{\frac{\max (K_{i,i} )+\min (K_{i,i} )}{2}} \end{aligned}$$
(41)
Thus, on Fig. 11, condition numbers are reduced drastically. However, the error still increases in case of XFEM. The error is clearly not only related to conditioning. Another explanation should be considered.
When looking at the XFEM discrete stiffness matrix, it is noticeable that the unknowns \(a_{2}\) and \(b_{2}\) obey to an almost similar equation (as \(H\Phi _2 \approx \Phi _2\)). The difference between those equations lies in the terms \(12\varepsilon \) and \(22\varepsilon \). Those terms imply the sum of heterogeneous quantities that leads to a tremendous loss of accuracy on the difference information \(2\varepsilon \). For instance, let us consider the following string of calculations in “double precision” arithmetic (8octet storage):
$$\begin{aligned} 1 (1 6.3847497084\mathrm{E}12)\;\approx 6.348\mathrm{E}12 \end{aligned}$$
(42)
The final result 6.348E–12, is quite different from the exact result 6.3487497084E–12: only four digits remain of the “difference information” between the equations. Hence ill conditioning indicates also a lesser accuracy within the assembled stiffness matrix due to roundoff errors and truncated information. Even highly effective preconditioners [21] cannot recover the truncated information. It is not surprising that, besides preconditioning, authors have used triple precision arithmetic to recover accuracy [24].
Remark
Because the error curves are shattered, the phenomenon might be also probabilistic and related to random roundoff errors in interaction with the solver algorithm. A comprehensive study with a wide range of solvers is presented (Figs. 12 and 13). This study shows that the type of solver interferes with the level of error, which makes numerical analysis on errors quite sensitive:

For iterative methods (PCG), the error is generally higher because those solvers are very sensitive to conditioning,

For matrix factorization based solvers (UMF), the error is lower than with iterative methods,

For matrix inversion based solvers, good results can be obtained frequently.
Numerical analysis of strongly discontinuous approximations with quadratic elements
The same onedimensional case is considered here. The same discretization of the weak form is used with quadratic shape functions. The results are plotted on Fig. 14.
For the three approximations, conditioning worsens very quickly, at about three times the rate of linear elements. More worrying is that the error increases. XFEM enrichment error worsens more quickly than the others.
On Fig. 15, the same preconditioners than in “Numerical behavior of strongly discontinuous approximations” section are considered. The wrong behavior noticed above still applies.
This numerical behavior with quadratic elements is well explained by the coupling between enrichment functions, as pictured on Fig. 16 with domain enrichment [33].
When \(measure(\left\{ {\Omega _{+} \cap Supp(\varPhi _S )} \right\} )\rightarrow 0\) for a given vertex node, the middle node satisfies likewise \(measure(\left\{ {\Omega _{+} \cap Supp(\varPhi _M )} \right\} )\rightarrow 0\), on the vertex patch. For such a configuration, the enrichment shape functions \(\chi _{\Omega +} \varPhi _S \) and \(\chi _{\Omega +} \varPhi _M \) become almost homothetic (Fig. 16).
Both functions can be related with the following expression,
$$\begin{aligned} \varPhi _S (\xi )=4\varPhi _M (\xi )+4(1\xi )^{2} \end{aligned}$$
(44)
when \(\xi \) goes to 1, the difference information between those functions decreases accordingly to \((1\xi )^{2}\). This generates a vanishing subspace (consisting of polynomials only defined on the small support \(\;\left\{ {\Omega _{+} \cap Supp(\varPhi _S )} \right\} \)), which implies a conditioning slope of at least 3 with quadratic shape functions (Fig. 15). In other words, the condition number \(\kappa (K)\) is bounded by:
$$\begin{aligned} \kappa (K)\ge C\varepsilon ^{3} \end{aligned}$$
(45)
Proof
Let \(\kappa (K)\) be the condition number of the stiffness matrix:
$$\begin{aligned} \kappa (K)=\frac{\max \left( \left\{ {e^{T}Ke} \right\} _{\left\ e \right\ _2 =1} \right) }{\min \left( \left\{ {e^{T}Ke} \right\} _{\left\ e \right\ _2 =1}\right) }\ge \frac{e_1^T Ke_1}{e_2^T Ke_2}\quad \forall e_1 ,e_2/\left\ {e_1} \right\ _2 =1,\left\ {e_2} \right\ _2 =1 \end{aligned}$$
(46)
where \(\left\ \right\ _2\) is the Euclidian norm 2.
In order to estimate the lower bound of the condition number, we consider the polynomials e
\(_1\) and e
\(_2\) illustrated on Fig. 17.
Those polynomials belong to the discrete space of [33] (likewise [7] and [26]). They can be used to determine a lower bound of the condition number. Functions \({e}_{1}\) and \({e}_{2}\) are expressed in the discrete space of [33] as:
$$\begin{aligned} \left\{ {{\begin{array}{l} {e_1 =\frac{1}{\sqrt{5}}\left( {2\Phi _1+\Phi _2 } \right) } \\ {e_2 =\frac{1}{\sqrt{17}}\left( {4\chi _{\Omega ^{}} \Phi _3 +\chi _{\Omega ^{}} \Phi _4 } \right) } \\ \end{array} }} \right. \end{aligned}$$
(47)
In the discrete space of [33], we have:
$$\begin{aligned} \left\ u \right\ _2 =\sqrt{\sum _{i\in I/I_H } {a_i^2 } +\sum _{j\in I_H } {\alpha _{j,1}^2 +\sum _{j\in I_H } {\alpha _{j,2}^2 } } } \end{aligned}$$
(48)
so that we have \(\left\ {e_1 } \right\ _2 =\left\ {e_2 } \right\ _2 =1\).
As \(\frac{1}{2}e_{1}^{T}Ke_1\) and \(\frac{1}{2}e_2^T Ke_2\) represent the elastic energy of these solutions, we have:
$$\begin{aligned} \frac{e_1^T Ke_1}{e_2^T Ke_2}=\frac{\int \nolimits _{0}^{1} {\left( {e_1^{\prime } } \right) ^{2}dx} }{\int \nolimits _{2\varepsilon }^{2} {\left( {e_2^{\prime } } \right) ^{2}dx} }=\frac{51}{80\varepsilon ^{3}} \end{aligned}$$
Thus,
$$\begin{aligned} \kappa (K)\ge C\varepsilon ^{3} \end{aligned}$$
The asymptotic behavior of \(\varPhi _M (\xi )\) [at the first order of \((1\xi )\)] is:
$$\begin{aligned} \varPhi _M (\xi )\approx \varPhi _S (\xi )/4 \end{aligned}$$
Then, the related d.o.f. become redundant, which leads to ill conditioning. As enrichment strategies are equivalent, the redundant d.o.f. pollutes also the approximation spaces of the other formulations [7, 26].
when \(\xi \) goes to 1, there is no coupling between Bernstein polynomials as observed with Lagrange polynomials. However, the polynomial \(\varPhi _S \) cancels out accordingly to \((1\xi )^{2}\), which also leads to the same conditioning slope of three (Fig. 18) (as shown in the previous section).
It is noticeable that [33] has better accuracy with Bernstein polynomials than with Lagrange polynomials (Figs. 18, 19). \(\square \)
Numerical behavior of singular approximations at the crack tip
Numerical analysis of crack approximations with linear elements: crack opening in mode 1
Given the capabilities of the software we used, only two approximations were tested here:
Other approximations are out of scope because,

Gupta et al. SGFEM kind of enrichment [25, 31] (let us recall that accordingly to [32] a genuine SGFEM enrichment is defined by the fact that the condition number of the associated stiffness matrix is of the same order—2 [38]—with respect to mesh refinement than the one of FEM) needs far too many additional Heaviside d.o.f. which are difficult to implement and may lead to conditioning issues,

Cutoff enrichment is not very convenient, as assembling global d.o.f. is out of the scope of the industrial software we used (Code_Aster). Moreover, the method does not extend properly in 3D [29].
Example 1
(Horizontal crack opening in mode I with linear elements). The domain geometry is a square defined by \(\Omega =\left[ {\,0.5,+\,0.5} \right] \times \left[ {\,0.5,+\,0.5} \right] .\) The meshing procedure subdivides the domain into regular sized cells, which edges are parallel to the crack. The size of the enrichment zone is \(r=0.1\).
The analytical solution in displacement corresponds to the exact mode I limited to the \(\surd r\) term in Williams series expansion [11] and is given by the first term of (15). Then, Dirichlet boundary conditions are applied on three sides. On the left side, Neumann conditions are preferred. As the left side is cut by the crack, Dirichlet conditions are more difficult to apply given the discontinuity of the displacement field (Fig. 20).
The condition number is estimated by MUMPS solver, so that the results shown here have to be taken only as rough estimates.
On Fig. 21, the condition number of XFEM almost skyrockets as noticed in [21]. We did not consider the preconditioner of [21] here to show how both enrichment strategies, scalar and vectorial, behave without expensive treatment. Even the conditioning slope of the vectorial enrichment is far from optimal. The expected optimal conditioning slope is two [25].
Then, the relative error in energy norm is computed accordingly to the following formula:
$$\begin{aligned} \Vert uu^{h}\Vert _\mathrm{energy}=\sqrt{\int \nolimits _\Omega {\left( {\sigma ^{h}\sigma } \right) :\left( {\varepsilon ^{h}\varepsilon } \right) }d\Omega \bigg /{\int \nolimits _\Omega {\sigma :\varepsilon }d\Omega }} \end{aligned}$$
(50)
On Fig. 22, the rate of convergence in energy norm of the XFEM scalar enrichment is underoptimal (around 0.91) as noticed in [17]. Optimality is recovered with the vectorial enrichment, the convergence rate being then 0.989. Nonetheless, XFEM is more accurate than the vectorial enrichment as predicted in Lemma 1.
Example 2
(Inclined crack opening in mode I with linear elements). We introduce a more realistic case with a chosen inclination of the crack at \(44.9{^{\circ }}\), in order to test the conditioning with both Heaviside and singular enrichments. The analytical solution is still the one of a mode I problem limited to the \(\surd r\) term in Williams series expansion given by the first term of (15) (Fig. 23). The inclined crack analytical solution is expressed through a rotation of \(44.9{^{\circ }}\) of the horizontal crack solution. The strain and stress tensors are rotated as well of the same angle. Then the same mix of Neumann and Diriclet boundary conditions are applied (Fig. 24), similarly to the case of the horizontal crack. We consider the same regular meshes as above which are not oriented accordingly to the crack surface here. The preconditioner of [21] is not applied here as in the previous case.
On Fig. 25, the conditioning of the XFEM vectorial enrichment becomes very high and reaches a steady value around \(10^{12}\). This behavior may be explained by the ill conditioning of the Heaviside enrichment. As the “Heaviside information” becomes very unbalanced, its ill conditioning overcomes the regular increase of conditioning expected with the vectorial enrichment (see Fig. 21) as explained in “Strong discontinuity approximation conditioning” section. Nonetheless, both enrichments have optimal convergence accuracy (see Fig. 26).
Numerical analysis of crack approximation with quadratic elements: crack opening in mode 1
Only the linear vectorial enrichment approximation [30] is working with a reasonable condition number. We recall that other approximations are not tested because,

XFEM conditioning skyrockets with quadratic elements, even with a small enrichment zone,

Gupta et al. SGFEM kind of enrichment [25, 31] needs far too many additional Heaviside d.o.f., so that it is difficult to implement and may lead to ill conditioning,

Cutoff with global d.o.f. assembling requires a “macro” element, which is not straightforward to implement in finite element software. Moreover the method does not extend properly in 3D [29].
Even if the linear vectorial enrichment converges at an optimal rate (in energy norm) with quadratic elements, the condition number is still very high (see Fig. 27). We did not use the preconditioner of [21], in order to test the conditioning of the vectorial enrichment. In “Singular approximation at a crack tip” section, a new enrichment strategy with better intrinsic numerical behavior, will be exposed.
Remark
a dedicated integration scheme is needed at the tip of the crack, to get an optimal rate of convergence in energy norm with quadratic elements [17]. Here, we used a GaussRadau integration rule of order 20 [39]. As the aim of the paper is not about singular integration, we did not try to optimize the procedure. Optimization of integration schemes has been well studied in [30, 40].