Mesh requirements for the finite element approximation of problems with sign-changing coefficients

Transmission problems with sign-changing coefficients occur in electromagnetic theory in the presence of negative materials surrounded by classical materials. For general geometries, establishing Fredholmness of these transmission problems is well-understood thanks to the \(\mathtt {T}\)-coercivity approach. Moreover, for a plane interface, there exist meshing rules that guarantee an optimal convergence rate for the finite element approximation. We propose here a new treatment at the corners of the interface which allows to design meshing rules for an arbitrary polygonal interface and then recover standard error estimates. This treatment relies on the use of simple geometrical transforms to define the meshes. Numerical results illustrate the importance of this new design.

1. From now on, we denote by \(\Vert \cdot \Vert _{\mathcal {O}} \) the \(L^2 \)-norm over the open set \(\mathcal {O} \).

2. It is an attempt only, because there is a mistake in the definition of the discrete operator \(\mathtt {T}_h\) that will be clarified in Sect. 5.

3. For instance, equality is obtained for \(\mathtt {R}_{\mathrm {new}} \) by choosing \(v_1 \) such that \(v^k_1 \in H^1_0(\Omega _1^k) \), for \(k=1,p\), with \(v_1^1 \) a symmetric function w.r.t. the line \(\theta =\frac{\beta }{2}\), and \(v_1^k(\rho ,\theta ) = v_1^k(\mathcal {R}_{k-1}(\rho ,\theta )) \), for \((\rho , \theta ) \in \Omega _1^k\), for \(k=2,p \). Similarly, equality for \(\mathtt {R'}_{\mathrm {new}} \) is obtained by choosing \(v_2 \) such that \(v^l_2 \in H^1_0(\Omega _2^l) \), for \(l=1,q\), with \(v_2^1 \) a symmetric function w.r.t. the line \(\theta =(p + \frac{1}{2})\beta \), and \(v_2^l(\rho ,\theta ) = v_2^1(\mathcal {R}_{l-1}(\rho ,\theta )) \), for \((\rho , \theta ) \in \Omega _2^l\), for \(l=2,q \).

4. With this choice \(u_r (\cdot , \theta )=0\) for \(\theta \in \lbrace n \beta , n \in \mathbb {Z}\rbrace \), then continuity is easily ensured at the crossing of the interfaces.

5. \(B_{2N} \) is composed of two trapezoids of side lengths \(s_{2N} \) and \(s_1\).

6. The strictly positive, upper limit distance up to which this property applies is a function of \(\min _{{p}} \rho _{min,{{p}}}\) and of \((\alpha _{{p}})_{{{p}}=1,N}\).

7. In accordance with the previous notations, one chooses \(\rho _{\min , p} \) such that \(\rho _{\min , p} < \rho _{c_p} \).

8. A Discontinuous Galerkin approach has been studied in [12].

9. Equality is obtained for \(p=q\) (corresponding to a plane interface) for which \(\Vert \mathtt {R}^{\mathrm {adm}}_i \Vert ^2 = \Vert \mathtt {R}^{{\mathrm {adm}}'}_i \Vert ^2 = 1 \), or \(p=1\): in that case there is only one admissible operator.

Authors and Affiliations

1. POEMS, ENSTA ParisTech, CNRS, INRIA, Université Paris-Saclay, 828 Bd des Maréchaux, 91762, Palaiseau Cedex, France

   Anne-Sophie Bonnet-Ben Dhia & Patrick Ciarlet Jr

2. Applied Mathematics Unit, School of Natural Sciences, University of California, Merced, 5200 North Lake Road, Merced, CA, 95343, USA

   Camille Carvalho

Corresponding author

Correspondence to Patrick Ciarlet Jr.

A Appendix

1.1 A.1 General construction of \(\mathtt {R} \) around corners

In this section we generalize the tilings method presented in Sect. 3 to any corner of angle \(\alpha \in 2\pi \mathbb {Q}\). Recall that we define \(\alpha = 2\pi \frac{p}{p+q} \) in \(\Omega _1\), with \(p ,q >0 \), \(p\ne q \) and \(p+q\) even. Proceeding as in Sect. 3, one builds admissible rotation- and symmetry-based operators \(\mathtt {R}^{{\mathrm {adm}}}\), and then take the average of these admissible operators to obtain the desired result (that is operators \(\mathtt {R} \) and \(\mathtt {R}'\) with the same minimal norm as in [2]). We propose \(\min (p,q) \) admissible operators below based on a simple algorithm.

Consider for instance that \(p < q \). One constructs p admissible operators \((\mathtt {R}^{\mathrm {adm}}_i)_{i=1,p} \) from \(V_1\) to \(V_2\), the i-th operator being obtained by (see Fig. 11 for an illustration):

1. 1)

   in \(\Omega _2^{q+1-k}\), \(k=1,i\): apply \(\mathtt {S}^1\) to \(v^k_1\);

2. 2)

   in \(\Omega _2^{p+1-k}\), \({k=i,p}\): apply \(\mathtt {S}^2\) to \(v^k_1 \);

3. 3)

   in \(\Omega _2^{l} \), \(l\in I:=\llbracket p+2-i, q-i\rrbracket \): from \(l=p+2-i\) to \(l=q-i\), apply \(\mathtt {R}_{i-(p+l)}\) to \(v^i_1\), then \(\mathtt {S}^2 \circ \mathtt {R}_{p+1-i-(l+1)} \) to \(v^i_1\), update \(l \rightarrow l+2\) and so on. In other words, alternatively apply a rotation-based operator and a rotation+symmetry-based operator to \(v^i_1\).

At step 3), \(|I|=q-p-1\) is odd since \(q+p\) is even, so one always finishes by \(\mathtt {R}_{2i-(p+q)}\) in \(\Omega _2^{q+1-i}\), which ensures continuity of \(\mathtt {R}^{\mathrm {adm}}_i v_1 \) on \(\partial \Omega _2^{q+1-i} \cap \partial \Omega _2^{q-i} \).

Scheme representing steps 1)-2)-3) for building \(\mathtt {R}^{\mathrm {adm}}_i \)

Full size image

One constructs p admissible operators \((\mathtt {R}^{{\mathrm {adm}}'}_i)_{i=1,p}\) from \(V_2\) to \(V_1\) similarly, the i-th operator being obtained by:

1. 4)

   in \(\Omega _1^{k}\), \({k=1,\min (i,p-1)}\): apply \(\mathtt {S}^1\) to \(v^{q+1-k}_2\);

2. 5)

   in \(\Omega _1^{k}\), \({k=\min (i+1,p),p}\): apply \(\mathtt {S}^2\) to \(v^{p+1-k}_2\);

3. 6)

   in \(\Omega _1^{i}\), if \(i<p\), add up the remaining contributions \((v^{l}_2)_{l \in {I'}}\), with \(I' := \llbracket p+1-i, q-i\rrbracket \): for \(l=p+1-i\) to \(l= q-i\), start by \(\mathtt {S}^2 \circ \mathtt {R}_{l-(p+1-i)} \) then apply \(-\mathtt {R}_{(l+1)+p-i}\), update \(l \rightarrow l+2\) and so on. If \(i=p\) one adds up \((v^{l}_2)_{l \in I'}\), where \(I'=\llbracket 2, q-p+1\rrbracket \): for \(l=2\) to \(l=q-p+1\) start with \( -\mathtt {R}_{l}\) then apply the rotation+symmetry-based operator \(\mathtt {S}^1 \circ \mathtt {R}_{(q -p +1)-(l+1)} \), update \(l \rightarrow l+2\) and so on.

At step 6), in both cases, \(|I'|=q-p\) is even since \(q+p\) is even, so one applies successively pairs of operators. Note also that at step 6), one adds contributions in the i-th pattern of \(\Omega _1 \), already considered at step 4) or 5), and remark that when the index is zero, \(\mathtt {R}_0 = \mathtt {I}\) so that the rotation+symmetry-based operator simply becomes a symmetry-based operator.

Since \(p+q\) is even, one can check that continuity is ensured at all interfaces of the patterns. The algorithm produces p operators from \(V_1\) to \(V_2\), respectively from \(V_2\) to \(V_1\). Let us give an example: for \(p=4\), \(q=6\) (that is \(\alpha = 4\pi /5 \)).

Then one finds four admissible operators from \(V_1\) to \(V_2 \) (each one corresponds to a column):

In this example, \(\vert I \vert =1 \) so one applies only one rotation-based operator (following step 3)).

Conversely, one finds four admissible operators from \(V_2\) to \(V_1 \):

Here \(\vert I' \vert =2\) so one applies two additional operators to \(v_2^l\), \(l \in I'\) (following step 6)).

With these guidelines one can write all operators \((\mathtt {R}^{{\mathrm {adm}}}_i)_{i=1,p} \), \((\mathtt {R}^{{\mathrm {adm}}'}_i)_{i=1,p} \) for any p, q such that \(p < q\) and \(p+q \) even. Note that by exchanging p with q (and \(\Omega _1\) with \(\Omega _2\)), one addresses similarly the case \(q< p\).

In the following, we set \( p < q\). As mentioned in Sect. 3, the problem is that, taken individually, no admissible operator \(\mathtt {R}^{\mathrm {adm}}_i\) (resp. \(\mathtt {R}^{{\mathrm {adm}}'}_i\)), \(i=1,p\), satisfies \(\Vert \mathtt {R}^{\mathrm {adm}}_i\Vert ^2 = I_\alpha \) (resp. \(\Vert \mathtt {R}^{{\mathrm {adm}}'}_i\Vert ^2 = I_\alpha \)), with \(I_\alpha \) defined in (9). Indeed, for all \(v_1 \in V_1 \), \(v_2 \in V_2 \), for \(i=1,p \), \(\mathtt {R}^{\mathrm {adm}}_i \) built from 1)-2)-3) and \(\mathtt {R}^{{\mathrm {adm}}'}_i \) built from 4)-5)-6), one gets the bounds

The bounds are sharp since:

• in the first case, one may choose \(v_1\ne 0\) such that \(v_1^k=0\) for \(k\ne i\): then \(\Vert \nabla (\mathtt {R}^{\mathrm {adm}}_i v_1) \Vert ^2_{\Omega _2} = (q-p +1) \, \Vert \nabla v_1 \Vert ^2_{\Omega _1}\) ;

• in the second case, given \(l\in I'(\ne \emptyset )\), one may choose \(v_2\ne 0\) such that \(v_2^k=0\) for \(k\ne l\): then \(\Vert \nabla (\mathtt {R}^{{\mathrm {adm}}'}_i v_2) \Vert ^2_{\Omega _1} = (q-p+1) \, \Vert \nabla v_2 \Vert ^2_{\Omega _2}\).

One can check that \(I_\alpha < q-p+1\) for all \(1<p<q \): \(I_\alpha - (q-p+1) = (p-q)(p-1)/p<0\)(^{Footnote 9}).

To get optimal operators (that is of norm equal to \( I_\alpha \)), one defines \(\mathtt {R} \), \(\mathtt {R}' \) as the average of the admissible operators:

Going back to our example with \(p=4\) and \(q =6\), one defines \(\mathtt {R}\) and \(\mathtt {R}'\) such that for all \(v_1 \in V_1\), \(v_2 \in V_2\)

These operators are optimal in the sense that \(\Vert \mathtt {R} \Vert ^2 = \Vert \mathtt {R}' \Vert ^2 = I_\alpha \). Proceeding as in Sect. 3, using the triangle inequality one finds that, for all \(v_1 \in V_1\),

By defining the matrix \(M = {\begin{pmatrix} 0 &{} 0 &{} 0 &{}1 \\ 0 &{} 0 &{} 3/4 &{} 1/4\\ 0 &{} 2/4 &{} 1/4&{} 1/4\\ 1/4 &{} 1/4 &{} 2/4&{} 0\\ 1/4 &{} 3/4 &{} 0&{} 0\\ 1 &{} 0 &{} 0 &{} 0 \end{pmatrix}} \), and \(\overrightarrow{W}:= (\Vert \nabla v^1_1 \Vert _{{\Omega }^1_{1}}, \Vert \nabla v^2_1 \Vert _{{\Omega }^2_{1}},\Vert \nabla v^3_1 \Vert _{{\Omega }^3_{1}} ,\Vert \nabla v^4_1 \Vert _{{\Omega }^4_{1}} )^{\intercal } \), one can check that \(\Vert \nabla (\mathtt {R} v_1) \Vert ^2_{\Omega _2} \le \Vert M \overrightarrow{W}\Vert ^2_2 \le \Vert M^\intercal M\Vert _2 \Vert \nabla v_1 \Vert ^2_{\Omega _1}\), leading to \(\Vert \mathtt {R} \Vert ^2 \le \Vert M^\intercal M\Vert _2\). Similarly, by defining \(M' = M^\intercal \), one can check that \(\Vert \mathtt {R}' \Vert ^2 \le \Vert {M'}^\intercal M'\Vert _2\).

Let us remark that the entries \(M_{lk}\), \(l=1,6\), \(k=1,4\), are such that \(M_{lk} = \sup \limits _{w_1 \in V_1} \displaystyle \frac{\Vert \nabla (\mathtt {R} w_1) \Vert _{\Omega _2^l}}{\Vert \nabla w_1^k \Vert _{\Omega _1^k}}\). For example for \(l = 4\), one has for all \(v_1 \in V_1\),

One obtains equality by taking, for every \(k = 1,4\), \(v_1\) such that \(v_1^i = 0\) for \(i \ne k\), which yields \(M_{41} = \frac{1}{4}\), \(M_{42} = \frac{1}{4}\), \(M_{43} = \frac{2}{4}\), \(M_{44} = 0\).

More generally, one obtains for instance for \(v_1 \in V_1 \)

Above, a sum over k appears due to the fact that for \(l=1,q\), the \((\mathtt {R}^{\mathrm {adm}}_i v_1)\vert _{\Omega _2^l} \) is a linear combination of isometry-based operators applied to some \(v_1^k\). Introducing \(M \in \mathcal {M}_{q,p}(\mathbb R)\) with entries \((M_{lk} )_{l=1,q,k=1,p}\), one has (see footnote 3 p. 9)

Let us give the general expression of the matrix M for any p, q, and evaluate \(\Vert M^{\intercal } M\Vert _2 \) to conclude. For \(n \in \llbracket 1,p\rrbracket \), define \(M_{p-n+1} \), \(\widetilde{M}_{p-n+1} \in \mathcal {M}_{p-n+1 , p} (\mathbb R)\) such that

where j denotes the column index. In the specific case where \(n=1\), the matrix \(M_p\) (resp. \(\widetilde{M}_p\)) is a square matrix for which the first column (resp. last column) has non zero entries. All entries are equal to 0 or \(\frac{1}{p}\), except on one diagonal where they range from \(\frac{n}{p}\) to \(\frac{p}{p} =1\). Then the shape of M depends on whether \(p\le \frac{q}{2}\) or not:

• if \(p \le q/2 \) : let \(m\in \mathbb N\) such that \(q = 2p + m\). Then the matrix M is written \(M = \begin{pmatrix} M_{p} \\ M'_m \\ \widetilde{M}_{p} \end{pmatrix} \) with \(M_p, \, \widetilde{M}_p \in \mathcal {M}_p(\mathbb R) \) (defined above for \(n=1\)) and respectively \(M'_m \in \mathcal {M}_{m,p}(\mathbb R) \) whose entries are all equal to 1 / p, with the convention that if \(m=0\), \(M'_m\) is empty. The example treated in Sect. 3 corresponds to that case with \(m=0\).

• if \(p > q/2 \) : let \(m\in \mathbb N^{{*}}\) such that \(q = 2p - m\). If \(m>2 \) then the matrix M is written \(M = \begin{pmatrix} M_{p-m+1} \\ M''_{m-2}\\ {\widetilde{M}_{p-m+1}} \end{pmatrix} \), with \(M_{p-m+1}\), \( {\widetilde{M}_{p-m+1} } \in \mathcal {M}_{p-m+1,p}(\mathbb R)\) (defined above for \(n=m\)) and resp. \({M}''_{m-2} \in \mathcal {M}_{m-2,p}(\mathbb R) \) such that (below i denotes the row index)

  

  It is convenient to define \(M''_{m-2}\) according to some \(d \in \mathbb {Z}\), defined such that \( p = 2 (m-2) + d\). For \(d > 1\), one can obtain some “central” columns with only \(\frac{1}{p}\) entries, whereas for \(d \le 1\), one gets central columns of the form \(( 0, \dots , 0,\frac{m-j}{p}, \frac{1}{p}, \dots , \frac{1}{p},\frac{1+i}{p}, 0, \dots , 0)^\intercal \) and \(( 0, \dots , 0,\frac{1+j}{p}, \frac{1}{p}, \dots ,\frac{1}{p}, \frac{m-i}{p}, 0, \dots , 0)^\intercal \) for some ad hoc i, j that govern the number of 0 entries. For example for \(d = 0\) the two central columns are given by \(( \frac{m-1}{p}, \frac{1}{p}, \dots , \frac{1}{p}, \frac{m-2}{p}, 0)^\intercal \), \(( 0,\frac{m-2}{p}, \frac{1}{p}, \dots , \frac{1}{p}, \frac{m-1}{p})^\intercal \) and for \(d = 1\) the central column is \(( \frac{m-1}{p}, \frac{1}{p}, \dots , \frac{1}{p}, \frac{m-1}{p})^\intercal \). The key is to start with the first and the last rows, then complete the rest accordingly. The matrix M simplifies to the matrix \(M = \begin{pmatrix} M_{p-1} \\ \frac{1}{p} \dots \frac{1}{p}\\ {\widetilde{M}_{p-1}} \end{pmatrix} \) when \(q=2p-1\), and to the matrix \(M = \begin{pmatrix} M_{p-1} \\ {\widetilde{M}_{p-1}} \end{pmatrix} \) when \(q=2p-2\). The previous example with \(p=4\), \(q=6\) corresponds to that last case.

Note that the cases \(p > q \) are obtained by taking the transpose of the matrix presented above.

Let us give two examples to illustrate the cases \(p\le q/2\) and \(p>q/2\).

Case \(p =2\), \(q =6\): in that case \(q = 2p +m\) with \(m =2\). Following 1)-2)-3) and using (33)-(34), one can check that for \(v_1 \in V_1\),

Case \(p =6\), \(q =8\): in that case \(q = 2p -m\) with \(m =4\). One can check that for \(v_1 \in V_1\),

Moreover, one checks easily by direct inspection the next result.

Proposition 1

The matrices \(M_{p-n+1}\in \mathcal {M}_{p-n+1,p}(\mathbb R)\), \(\widetilde{M}_{p-n+1}\in \mathcal {M}_{p-n+1,p}(\mathbb R)\), \({M}'_{m}\in \mathcal {M}_{m,p}(\mathbb R)\), \({M}''_{m-2}\in \mathcal {M}_{m-2,p}(\mathbb R)\) satisfy the following properties:

1. 1)

   \(\sum \limits _{l=1}^{p-n+1} ({M}_{p-n+1})_{lk} = {\left\{ \begin{array}{ll} 0 &{}\quad \text {if }k< n \\ \\ \frac{2k-n}{p} &{}\quad \text {if } k \ge n \end{array}\right. } \) ; \(\quad \forall l = 1, p-n+1 , \, \sum \limits _{k=1}^{p} ({M}_{p-n+1})_{lk} = 1\);

2. 2)

   \( \quad \sum \limits _{l=1}^{p-n+1} (\widetilde{M}_{p-n+1})_{lk} = {\left\{ \begin{array}{ll} \frac{2(p+1-k)-n}{p} &{}\quad \text {if }k \le p-n+1\\ \\ 0 &{}\quad \text {if } k > p-n+1 \end{array}\right. } \) ; \( \quad \forall l = 1, p-n+1 , \, \sum \limits _{k=1}^{p} (\widetilde{M}_{p-n+1})_{lk} = 1\);

3. 3)

   \(\forall k=1, p, \, \sum \limits _{l=1}^{m} ({M}'_{m})_{lk} = \frac{m}{p}\); \(\quad \forall l = 1, m , \, \sum \limits _{k=1}^{p} ({M}'_{m})_{lk} = 1\);

4. 4)

   \(\sum \limits _{l=1}^{m-2} ({M}''_{m-2})_{lk} = {\left\{ \begin{array}{ll} \frac{2(k-1)}{p} &{}\quad \text {if } k< \min (m, m-2+d)\\ \\ \frac{m-2}{p} &{}\quad \text {if } m\le k \le {p-m}+1 \quad \text{ for } d>1 \\ \\ \frac{q}{p} &{}\quad \text {if } m-2+d \le k \le p-m+3-d\quad \text{ for } d \le 1 \\ \\ \frac{2(p-k)}{p} &{}\quad \text {if } \max ({p-m}+1, p-m+3-d) < k \end{array}\right. } \) ; \(\quad \forall l = 1, {m-2} , \, \sum \limits _{k=1}^{p} ({M}''_{m-2})_{lk} = 1\).

Proposition 2

The matrix \({M\in \mathcal {M}_{q,p}(\mathbb R)}\) satisfies the following properties:

Proof

Using proposition 1 one can deduce all the above results. Let us simply detail how to obtain the first result of [3], i.e. when \(q\ge p\). First consider the case where \(q = 2 p +m\), \(m \in \llbracket 1,p\rrbracket \). Then for all \(k = 1,p\) one has

Now let us consider the case \(q = 2 p -m\), \(m \in \llbracket 3, p -1\rrbracket \) and suppose that \(d >1\): then \(\min (m, m-2+d) = m\) and \(\max ({p-m}+1, p-m+3-d) = p-m+1\). Then for all \(k = 1,p\)

One reaches the same result in the specific cases \(d \le 1\) by noticing that for \(l=1,p-m+1\), \((M_{p-m+1})_{lk} =(\widetilde{M}_{p-m+1})_{lk} =0\) for \(k = m-2+d,m-1\). \(\square \)

It follows that

Proposition 3

For all \({p, \, q >0 }\) let \({M\in \mathcal {M}_{q,p}(\mathbb R)}\) be a matrix which satisfies the properties of proposition 2. Then

Proof

Define \(A:=M^{{\intercal }} M \), it holds \( \Vert A \Vert _2 = \lambda _{\max } = \max \limits _{\lambda \in \sigma (A)}\lambda (A){>0}\), where \(\sigma (A) \) is the set of eigenvalues of A. Using Gershgorin circle theorem, one bounds the spectrum of A as follows:

then

Due to the second and the third properties in proposition 2 satisfied by M, one finds

Let us prove that \(\lambda _{\max } = \max \left( \frac{p}{q}, \frac{q}{p}\right) \). Consider the vector \(\vec {W} \in \mathbb {R}^p\) such that \(\vec {W} = (1, \dots , 1)^\intercal \). Suppose for instance that \(p \le q\), then using again the second and the third properties in proposition 2, we get for all \(j = 1,p\)

namely \((M^\intercal M) \vec {W}= \max \left( \frac{p}{q}, \frac{q}{p}\right) \vec {W}\). \(\square \)

Consequently, the proposed operator \(\mathtt {R} \) in (33) is of optimal norm. One proceeds similarly for \( \mathtt {R}'\) by considering \(M^\intercal \) instead of M as the roles of p and q are exchanged:

Remark 4

There is always a unit entry in M according to proposition 2 (denoted by \(M_{k_0l_0}\)), one readily checks that if \(w_1^0\in V_1\) with \(\text{ supp }(w_1^0)\subset \Omega _1^{k_0}\), then it follows \( \Vert \nabla (\mathtt {R} w_1^0)\Vert ^2_{\Omega _2 }\ge \Vert \nabla w_1^0\Vert ^2_{\Omega _1 }\). Hence \(\Vert \mathtt {R} \Vert ^2\ge 1\). Similarly, \(\Vert \mathtt {R}' \Vert ^2\ge 1\).

1.2 A.2 Weighted estimates for operators \(\mathtt {R}\)

In §A.1 we provided bounds for the norms of the geometry-based operators \(\mathtt {R} \). Here we provide bounds for the operator norm when we use a localization process (see Sect. 4), that is when the operator \(\mathtt {R}\) is locally applied in the neighborhood of the interface thanks to a cut-off function \(\xi \) (defined as in Sect. 4) whose support is localized either near a corner (proposition 4) or a straight line (proposition 5) of the interface \(\Sigma \). We use the same notations as in §A.1.

Proposition 4

Let \(\xi \) be a smooth positive function with support S, that depends only on the distance to the corner of angle \(\alpha = 2\pi \frac{p}{p+q}\). Then

where \(I_\alpha = \max \displaystyle \left( \frac{p}{q}, \frac{q}{p}\right) \), and \(\mathtt {R} \) is a rotation- and symmetry-based operator from \(V_1\) to \(V_2\) defined as in (21).

Proof

Let \(w_1 \in V_1 \).

Then similar to (34) with the change of variables \((\rho , \theta ) \mapsto (r_k,\varphi _k)\) for \(k=1,p \) that maps \(\Omega _2^l \) to \(\Omega _1^k\) (note that \(\xi (r_k) = \xi (\rho )\)) one finds

Introducing \(\overrightarrow{W}_\xi =( \Vert \xi ^{1/2} \nabla w^1_1 \Vert _{\Omega ^1_1 \cap S}, \dots ,\Vert \xi ^{1/2} \nabla w^p_1 \Vert _{\Omega ^p_1 \cap S} )^\intercal \), then one has \(M \overrightarrow{W}_\xi = \sum \nolimits _{k=1}^{p} M_{lk } \Vert \xi ^{1/2} \nabla w^k_1 \Vert _{\Omega ^k_1 \cap S} \), using proposition 3 one finally gets

\(\square \)

Remark 5

Following remark 4, one can find \(w_1^0\in V_1\) such that \( \Vert \xi ^{1/2}\,\nabla (\mathtt {R} w_1^0)\Vert ^2_{\Omega _2 \cap S }\ge \Vert \xi ^{1/2}\,\nabla w_1^0\Vert ^2_{\Omega _1 \cap S}.\)

Proposition 5

Let \(\xi \) be a smooth positive function with support S that is symmetric with respect to the interface. Then for all \(w_1 \in V_1 \)

where \(\mathtt {R}\) is the symmetry-based operator (22).

Proof

This inequality is obtained using the change of variables \((x^\Sigma ,y^\Sigma )\rightarrow (x^\Sigma ,-y^\Sigma )\) in \(\Omega _2 \cap S\). For all \( w_1 \in V_1\)

\(\square \)

1.3 A.3 Local and global interpolation estimates

Let \({\hat{\tau }}\) be the reference triangle, with corners (1, 0), (0, 1) and (0, 0), and let \({\hat{I}}_d\) be the Lagrange interpolation operator over \(\mathbb {P}_d({\hat{\tau }})\). Given \((\mathcal {T}_h)_h\) a regular family of triangulations of a domain \({\Omega }\) we call \(I_\tau ^d\) the Lagrange interpolation operator over \(\mathbb {P}_d(\tau )\), for all h and all \(\tau \in \mathcal {T}_h\).

Lemma 4

On the reference triangle \({\hat{\tau }}\), one has the estimate

As a consequence, one has the uniform local estimate

Proof

Write \({\hat{\chi }}({\hat{x}},{\hat{y}}) = \alpha + {\hat{\chi }}_1({\hat{x}},{\hat{y}})\), resp. \({\hat{v}}({\hat{x}},{\hat{y}}) = {\hat{v}}_-({\hat{x}},{\hat{y}}) + {\hat{v}}_d({\hat{x}},{\hat{y}})\) where \({\hat{\chi }}_1({\hat{x}},{\hat{y}})= \beta {\hat{x}} + \gamma {\hat{y}}\), resp. \({\hat{v}}_d({\hat{x}},{\hat{y}})=\sum _{i=0,d}a_i {\hat{x}}^i{\hat{y}}^{d-i}\) and \(deg({\hat{v}}_-)<d\). For \(i=0,d\), define \({\hat{ e}}_i({\hat{x}},{\hat{y}})= {\hat{x}}^{i+1}{\hat{y}}^{d-i}-{\hat{I}}_d({\hat{x}}^{i+1}{\hat{y}}^{d-i})\), resp. \({\hat{f}}_i({\hat{x}},{\hat{y}})= {\hat{x}}^{i}{\hat{y}}^{d+1-i}-{\hat{I}}_d({\hat{x}}^{i}{\hat{y}}^{d+1-i})\). Note that \(|{\hat{\chi }}|_{W^{1,\infty }({\hat{\tau }})}=\max (|\beta |,|\gamma |)\).

Since there holds \({\hat{\chi }}{\hat{v}} - {\hat{I}}_d({\hat{\chi }}{\hat{v}}) = {\hat{\chi }}_1{\hat{v}}_d - {\hat{I}}_d({\hat{\chi }}_1{\hat{v}}_d)=\beta \sum _{i=0,d}a_i{\hat{e}}_i({\hat{x}},{\hat{y}})+\gamma \sum _{i=0,d}a_i{\hat{f}}_i({\hat{x}},{\hat{y}})\), one finds

It follows that \(\Vert {\hat{\nabla }}({\hat{\chi }}{\hat{v}} - {\hat{I}}_d({\hat{\chi }}{\hat{v}}))\Vert _{{\hat{\tau }}}\le {\hat{C}}_d\, |{\hat{\chi }}|_{W^{1,\infty }({\hat{\tau }})}(\sum _{i=0,d}|a_i|^2)^{1/2}\), with \({\hat{C}}_d =\sqrt{2} [\sum _{i=0,d}(\Vert {\hat{\nabla }} {\hat{e}}_i\Vert _{{\hat{\tau }}}^2+\Vert {\hat{\nabla }} {\hat{f}}_i\Vert _{{\hat{\tau }}}^2)]^{1/2}\).

Then, we remark that the \(\ell ^2\)-norm of the coefficients is a norm over \(\mathbb {P}_d({\hat{\tau }})\), hence it is also a norm over its vector subspace \(\mathbb {P}_d^{zmv}({\hat{\tau }})\) made of zero-mean value polynomials on \({\hat{\tau }}\). Now, over \(\mathbb {P}_d^{zmv}({\hat{\tau }})\), the semi-norm \(\Vert {\hat{\nabla }}\cdot \Vert _{{\hat{\tau }}}\) is also a norm and because \(\mathbb {P}_d^{zmv}({\hat{\tau }})\) is a finite dimensional vector space, both norms are equivalent. Noting finally that, starting from \({\hat{v}}\), one has only to modify the degree-0 coefficient to obtain a zero-mean value polynomial, we finally get that there exists \({\hat{C}}'\) independent of \({\hat{v}}\) such that \((\sum _{i=0,d}|a_i|^2)^{1/2}\le {\hat{C}}' \Vert {\hat{\nabla }}{\hat{v}}\Vert _{{\hat{\tau }}}\). Taking \({\hat{C}}={\hat{C}}_d\,{\hat{C}}'\) leads to (35).

We recall that provided the family of triangulations \((\mathcal {T}_h)_h\) is regular, there exists \(\mathtt {s}>0\) such that, for all h and for all \(\tau \in \mathcal {T}_h\), there holds \(h_\tau \le \mathtt {s}\,\rho _\tau \), where \(\rho _\tau \) is the diameter of the largest ball that can be inscribed in \(\tau \). One can then derive (36) from (35) by using the affine mapping from the reference triangle \({\hat{\tau }}\) to the triangle \(\tau \in \mathcal {T}_h\). We refer for instance to [16]. We report here the computations for the sake of completeness (\(C_0,C_1,\cdots \) are constants that are independent of h and \(\tau \)):

that is (36) with \(C=C_2\,C_3\). \(\square \)

Let us prove the estimate over \(\Omega _2^h=int(\bigcup _{p=1,P}\cup _{\tau \in \mathcal {T}^{p}_{h,2}}\tau )\). Recall that \(V := H^1_0(\Omega ) \), \(V^h_{(d)} := \lbrace v \in V \, : \, v\vert _\tau \in \mathbb {P}_d(\tau ), \, \forall \tau \in \mathcal {T}_h\rbrace \), and \(I_h\) is the interpolation operator on \(V^h_{(d)} \).

Lemma 5

Consider a cut-off function \(\chi \), and denote by \(\chi _h \) its interpolation of degree 1. For all \(v^h \in V^h_{(d)} \),

Proof

To obtain (37), we evaluate the \(L^2\)-norm of \(\nabla (\chi _{{h}}v^h-I_h(\chi _{{h}}v^h))\) on \(\Omega _2^h\) by splitting \(\Omega _2^h\) into triangles, and then going back to the reference triangle to use the uniform estimate (36):

Using the definition of the meshsize h yields

One concludes using the stability estimate (25). \(\square \)

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