Dinh-Van Nguyen
email: dinh-van.nguyen@inria.fr

Fawzi Nashashibi

Trung-Kien Dao
email: trung-kien.dao@mica.edu.vn

Eric Castelli

Improving Poor GPS Area Localization for Intelligent Vehicles

Keywords: GPS, Laser, SLAM, WiFi Fingerprinting, Fusion, Localization, Intelligent Vehicle, Autonomous Vehicle, Particle Filter

Precise positioning plays a key role in successful navigation of autonomous vehicles. A fusion architecture of Global

Positioning System (GPS) and Laser-SLAM (Simultaneous Localization and Mapping) is widely adopted. While Laser-SLAM is known for its highly accurate localization, GPS is still required to overcome accumulated error and give SLAM a required reference coordinate. However, there are multiple cases where GPS signal quality is too low or not available such as in multi-story parking, tunnel or urban area due to multipath propagation issue etc. This paper proposes an alternative approach for these areas with WiFi Fingerprinting technique to replace GPS. Result obtained from WiFi Fingerprinting will then be fused with Laser-SLAM to maintain the general architecture, allow seamless adaptation of vehicle to the environment.

I. INTRODUCTION

A fusion system of Laser-SLAM and GPS is commonly adopted for localization of smart vehicle. Since Laser-SLAM is only capable of delivering accurate localization result within its local coordinate system, GPS information is required to map SLAM coordinate to a global one. Also, Laser-SLAM motion model is prone to accumulated error after a long run. Thus a combination with global GPS position will not only help to improve positioning results but also potentially solve SLAM problem of loop closure [START_REF] Trehard | Credibilist simultaneous Localization and Mapping with a LIDAR[END_REF], [START_REF] Levinson | Map-Based Precision Vehicle Localization in Urban Environments[END_REF], [START_REF] Rezaei | Kalman Filter-Based Integration of DGPS and Vehicle Sensors for Localization[END_REF]. Still, there are areas with low to no GPS signal such as Multi-story parking, tunnel or urban area with dense construction. Hence, it is necessary to find an alternative approach for these areas.

A lot of studies recently address this issue with different approaches. Studies in [START_REF] Levinson | Map-Based Precision Vehicle Localization in Urban Environments[END_REF], [START_REF] Wahl | Localization inside a populated parking garage by using particle filters with a map of the static environment[END_REF], [START_REF] Fouque | Enhancement of Global Vehicle Localization using Navigable Road Maps and Dead-Reckoning[END_REF] make use of environment static map to improve SLAM matching confidence. A network of cooperating vehicles is explored in [START_REF] Parker | Vehicle Localization in Vehicular Networks[END_REF], [START_REF] Drawil | Intervehicle-Communication-Assisted Localization[END_REF] with the aim to improve localization of each vehicle. A complete solution utilizes a wide range of information such as GPS, radio frequency identification, vehicle to vehicle and vehicle to infrastructure communication is introduced in [START_REF] Amini | Improving GPS-Based Vehicle Positioning for Intelligent Transportation Systems[END_REF]. However, these solutions often fall into the problem of fusing different sensors data and information. Dealing with various form of information standards, errors and uncertainties bring unstable to the system as a whole. Moreover, the combination of GPS and SLAM performs well in most of the cases, a complicated addition to the system for cases like tunnel, car park should be avoided.

This paper proposes to use WiFi Fingerprinting techniques as a replacement for GPS information in weak GPS area. WiFi Fingerprinting localization is a technique where WiFi map of targeted environment will be learned in training phase. A prediction phase will carry out location by comparing current signal with pre-learned radio map. While learning radio map of the targeted area is mandatory for WiFi Fingerprinting, the effort of collecting such data is much less in comparison to map-based and camera methods. Moreover, WiFi Fingerprinting is capable of mimicking GPS behavior in the fusion solution with Laser-SLAM. The fusion system of GPS and SLAM can be smoothly switched to WiFi and SLAM when certain conditions are met. This reduces uncertainty need to be added to the vehicle.

The main idea of this study is to replace poor GPS signal in certain areas with results from WiFi Fingerprinting localization method. A fusion strategy using a bootstrap particles filter of GPS -SLAM and WiFi -SLAM is also proposed. This fusion approach will help vehicles to adapt seamlessly to the change of environment. This paper structure is as follows. In section II, WiFi Fingerprinting method using ensemble neural network is explained together with a fusion strategy. Section III describes experiments conducted. Finally, Section IV concludes the paper with expected future improvement.

II. METHODOLOGY

A. WiFi Fingerprinting method

WiFi fingerprinting localization is a technique based on learning the map of WiFi RSSI (Radio Signal Strength Indicator) available in the environment at multiple reference points spread across the environment. The main assumption is that each reference point has a unique pattern of RSSIs of all available Access Points (APs). This pattern then allows vehicles to recognize the location just by scanning RSSIs for next visit. This method has two main steps. The first step is a training phase with multiples WiFi scanning at each reference position in an environment is recorded together with its coordinate. The second step is a prediction phase where scanning data of RSSIs without coordinate is compared to data registered in step 1. A prediction from the second step is likely the current position of the vehicle.

The major challenge in this approach is RSSIs of standard WiFi system are often noisy due to interference, and multipath propagation problem. A raw data processing will be performed on noisy and unstable wifi signal strength. Upon recording a vector of RSSI and the corresponding location as in [START_REF] Trehard | Credibilist simultaneous Localization and Mapping with a LIDAR[END_REF] where 𝑥 𝑖,𝑗 is WiFi RSSI from jth WiFi APs recorded in ith scan, 𝜌 𝑙 is a label which has corresponding coordinate at position of sampling and n is fixed constant. Here, n should be greater than total number of APs in learning environment

{𝑥 𝑖,1 , 𝑥 𝑖,2 , 𝑥 𝑖,3 , … , 𝑥 𝑖,𝑛 , 𝜌 𝑙 } (1)
Collected data will be normalized in the range of [-1, 1) where in particular scan, detected AP RSSI would be normalized (2) in the range [0, 1) with 0 as weakest possible signal strength and 1 as strongest possible signal strength. Other undetected APs at 𝜌 𝑙 will take value -1

𝑥 𝑖 = { -1, 𝐴𝑃 𝑖 𝑢𝑛𝑑𝑒𝑡𝑒𝑐𝑡𝑒𝑑 1 - (-1)×𝑅𝑆𝑆𝐼 100 , 𝐴𝑃 𝑖 𝑑𝑒𝑡𝑒𝑐𝑡𝑒𝑑 (2)
A system of 2 omnidirectional wifi antenna is mounted close to each other to minimize the impact of signal interference or multipath problem of the radio signal. In a particular scan, the antenna with highest RSSI between the two will be recorded. This is due to the observation that interference and multipath propagation will most likely reduce received signal strength. Thus, the higher RSSI will likely to be closer to direct signal without interference.

For the second step, A set of neural networks is implemented to learn from training data and perform. However, as RSSI appears to be noisy and scanning frequency of WiFi is low in comparison to movement speed, data will be considered to be a high variant. Using a method called Ensemble Bagging (Bootstrap Aggregating) Neural Network, which is well-known for combining multiple learning models to derive better results of prediction [START_REF] Breiman | Bagging Predictors[END_REF], [START_REF] Dietterich | Ensemble Methods in Machine Learning[END_REF], the system is expected to overcome high variant and noisy data issue.

Consider a classification method with a pair {𝑋 𝑖 , 𝑌 𝑗 } where 𝑋 𝑖 is a vector of predictor variable and 𝑌 𝑗 denotes a response, 𝑌 𝑗 ∈ {1, 2, . . 𝑚} . The target function is 𝑃(𝑌 = 𝑗 |𝑋 = 𝑥) for classification. A function estimator which results from a set of training samples and a classification model is formed [START_REF] Rezaei | Kalman Filter-Based Integration of DGPS and Vehicle Sensors for Localization[END_REF].

𝑔(•) = ℎ((𝑋 1 , 𝑌 1 ), (𝑋 2 , 𝑌 2 ), … , (𝑋 𝑛 , 𝑌 𝑚 ) )) (3)
Bagging algorithm consists following steps:

Step 1: Construct a bootstrap sample (4) by randomly sampling with replacement n times from original data:

(𝑋 ^1, 𝑌 ^1), (𝑋 ^2, 𝑌 ^2), … , (𝑋 ^n, 𝑌 ^m) (4)
Step 2: Compute bootstrapped estimator 𝑔 ^(•) in ( 5) by applying same classification model to newly formed bootstrap sample.

𝑔 ^(•) = ℎ((𝑋 ^1, 𝑌 ^1), (𝑋 ^2, 𝑌 ^2), … , (𝑋 ^n, 𝑌 ^m)))

(5)

Step 3: Repeat two steps above for K times with K is large. The bagging estimator is [START_REF] Parker | Vehicle Localization in Vehicular Networks[END_REF].

𝑔 ^𝑏𝑎𝑔𝑔 (•) = 1 𝐾 (∑ 𝑔 ^𝑖(•) 𝐾 𝑖=1 ) 
Theoretically, the bagging estimator is [START_REF] Drawil | Intervehicle-Communication-Assisted Localization[END_REF] as K goes to infinity:

𝑔 ^𝑏𝑎𝑔𝑔 (•) = 𝐄 ^[𝑔 ^(•)] (7)
In practice, a finite large K is expected to improve the accuracy of Monte Carlo approximation. In this study, a model of the neural network is constructed without carefully tuning parameters. Then K is chosen at 100 simple neural networks for ensemble purpose.

B. Fusion of WiFi Fingerprinting with Laser-SLAM

As mentioned in section I, this study aims to maintain the architecture of fusion between GPS -SLAM for a seamless adaptation of environmental conditions. A fusion strategy using particle filter is applied to accomplish the goal. The general design is demonstrated in Fig. 1. In this solution, GPS signal / WiFi localization result with expected error will serve as variance σ in covariance of particle estimation and WiFi/ GPS location. Thus, taking WiFi/ GPS location as median (𝜇 𝑊𝑖𝐹𝑖 and 𝜇 𝐺𝑃𝑆 respectively) it is possible to update the score of each particle using Gaussian distribution [START_REF] Amini | Improving GPS-Based Vehicle Positioning for Intelligent Transportation Systems[END_REF]:

𝑃(𝑋 𝑖,𝑡 | 𝜇 𝑡 , σ 𝑡 ) = 1 √2σ 𝑡 2 𝜋 𝑒 -(𝑥-𝜇 𝑡 ) 2 2σ 𝑡 2 ⁄ (8)
Here, 𝜇 𝑡 and σ 𝑡 is determined by quality of WiFi / GPS location estimation and 𝑋 𝑖,𝑡 is estimated location of particle i at current time t. While Dilution of precision (DOP) of GPS can be used to estimate these two variables when GPS is available, the corresponding values for WiFi are fixed and estimated through evaluation of empirical experiment results. This fusion architecture is interesting since it allows the system to switch between GPS signal and WiFi estimation effortlessly. Since WiFi Fingerprinting has constant estimated error, switching between GPS and WiFi is decided by comparing quality of available GPS to WiFi estimation. Hence, a stable result is expected from the fusion system.

𝑃(𝑋 𝑖,𝑡 | s

𝑡 ) = 𝑃(𝑋 𝑖,𝑡 | 𝜇 𝑡 , σ 𝑡 ) 𝑆(𝑋 𝑖,𝑡 ) (9)
𝐸 𝑡 = ∑ 𝑃(𝑋 𝑖,𝑡 | s 𝑡 ) 𝑋 𝑖,𝑡 𝑖=0 𝑛 [START_REF] Dietterich | Ensemble Methods in Machine Learning[END_REF] With particle evolution model using odometer sensors data and covariance updating from WiFi/GPS, the estimated particles will represent results from traditional SLAM motion model component. A multinomial resampling [11], [START_REF] Salmond | Novel approach to nonlinear/non-Gaussian Bayesian state estimation[END_REF] of particles is required at this stage to refine particles pool. Significant particles will be brought to matching and scoring step with laser data in the current position. Equation ( 9) shows how SLAM score 𝑆(𝑋 𝑖,𝑡 ) for matching process of each particle will be fused with score from WiFi/ GPS covariance correction. By using WiFi/ GPS expected errors as covariance of fusion, particles will tend to be distributed around WiFi/GPS reference. Finally, an estimation of current position is made by normalize score all particles and take mean value as shown in [START_REF] Dietterich | Ensemble Methods in Machine Learning[END_REF].

III. EXPERIMENTS AND RESULTS

Experiments are carried out in INRIA Rocquencourt campus using a cyber car and a version of Credibilist SLAM [START_REF] Trehard | Credibilist simultaneous Localization and Mapping with a LIDAR[END_REF]. The cyber car (Fig. 3) is equipped with one front IbeoLux LIDAR sensor, a standard 2.4 GHz WiFi antennas for WiFi Fingerprinting and an IMU for odometer information. A Real-Time Kinematic GPS (RTK GPS) antenna is mounted on the car to give precise ground truth.

The cybercar is then guided through test paths as shown in (2): A combination of WiFi/standard GPS and Credibilist SLAM with WiFi and SLAM for A-B, B-A; Standard GPS and SLAM for B-C-D-B. Here, the path from A-B is simulated for poor-GPS case where WiFi Fingerprinting is available. The system is then tested for the ability to replace GPS with WiFi information and vice versa.

Before integrating WiFi into the system, it is necessary to investigate the characteristic and expected error bound of WiFi alone as a localization method. This is done by another experiment along the entire test path A-B. For this path, a radio map is learned including 15 reference points. At each point, 30 scans of WiFi signal are collected as training data. An ensemble of 50 neural networks, each with 170 input neurons, 90 nodes at hidden layer and 15 outputs are trained. Average of predictions from all networks will then be calculated. A threshold of 0.55 is set for which prior probability from prediction must overcome to be counted as a valid localization estimate. Fig. 4 shows an independent WiFi localization result with particle filter. It proves that alone, WiFi localization is able to track vehicle. The Euclidian error of each localization output is calculated and presented in Fig. 5. The average error in entire path is 3.328 meters and 98% of errors are under 6 meters. This allows us to set σ 𝑊𝑖𝐹𝑖 at 6 meters for fusion equation. In the first scenario, GPS information is assumed to be available for whole path A-B-C-D-B-A. Fig. 6 shows a RTK GPS ground truth in green and a fusion of SLAM and standard GPS in blue. With assists from GPS, SLAM can recognize the previous location and follow the vehicle with only one LIDAR sensor setup. In the second scenario, the path from A-B is assumed to be a weak GPS area. The vehicle is required to activate WiFi localization for A-B then replace it with GPS when possible (B-C-D). Localization result is shown in Fig. 7 with the green ground truth of RTK GPS and the blue fusion localization result of WiFi, GPS and SLAM. In this case, the average error of fusion localization method is estimated at 2.7 meters with maximum error of 8 meters.

IV. CONCLUSION

This paper presents an alternative solution for weak GPS area with WiFi Fingerprinting localization technique. A fusion strategy for WiFi, GPS and SLAM are proposed to adapt the system to the change of environmental conditions seamlessly. Early results show that a combination of WiFi and SLAM can be a replacement for weak GPS and SLAM fusion. In the future, several techniques will be applied to improve the system such as: a deep learning strategy for wifi fingerprinting as well as a multi-receiver setup for wifi fingerprints.

V. ACKNOWLEDGMENTS

Authors express their gratitude to the French project VALET and the RITS Team for its support in the development of this work.

Fig. 1 .

 1 Fig. 1. Fusion architecture of WiFi/ GPS and Laser SLAM

Fig. 4 .

 4 There are four intersections noted on the map: A, B, C, D. Test paths are sequenced as follows: A -B -C -D -B -A. Path from A -B with sufficient WiFi infrastructure will be trained for WiFi Fingerprinting localization. A standard GPS with expected error of 6 meters is utilized in combination with SLAM. There are 2 experimental scenarios: (1): A fusion of standard GPS and Credibilist SLAM path: A-B-C-D-B-A.

Fig. 2 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .

 24567 Fig. 2. Experiment environment -INRIA Rocquencourt campus

Fig. 8 .

 8 Fig. 8. Fusion of WiFi, Standard GPS and SLAM (σ 𝐺𝑃𝑆 = 6, σ 𝑊𝑖𝐹𝑖 = 6)


Anne-Sophie Bonnet-Ben Dhia

Camille Carvalho

Anne-Sophie Bonnet

Ben Dhia

Patrick Ciarlet

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

Keywords: T-coercivity, transmission problem, sign-changing coefficient, conforming finite element method, T-conforming mesh

come

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

Introduction and setting of the transmission problem with signchanging coefficients

Our aim is to solve the problem Find u ∈ H 1 0 (Ω) such that:

-div(σ∇u) = f in Ω, (1)
where Ω ⊂ R 2 is a bounded domain partitioned into two regions, σ is a scalar, real-valued, sign-changing coefficient, and f is some given data. In electromagnetic theory, this problem can be interpreted as a transmission problem, in a domain composed of a classical dielectric material (σ > 0), and a negative material (σ < 0). A negative material can be for example a metal at optical frequencies, or a metamaterial (e.g. [START_REF] Smith | Metamaterials and negative refractive index[END_REF][START_REF] Ramakrishna | Physics of negative refractive index materials[END_REF]), for which some physical parameters become negative (the permittivity for metals, both the permittivity and the permeability for metamaterials). Due to the sign-changing coefficient σ, well-posedness of problem (1) is not guaranteed. In particular, classical tools such as Lax-Milgram theorem do not apply since the coercivity on H 1 0 (Ω) × H 1 0 (Ω) of the corresponding sesquilinear form

a : (v, w) → ˆΩ σ∇v • ∇w dΩ, (2)
is lost. However, over the past decade, techniques have been developed to establish well-posedness, under appropriate conditions, via the T-coercivity theory: introduced in [START_REF] Bonnet-Ben Dhia | Time harmonic wave diffraction problems in materials with sign-shifting coefficients[END_REF], it consists in building isomorphisms T such that the form (v, w) → a(v, Tw) is coercive on H 1 0 (Ω) × H 1 0 (Ω). For short, we say in this case that a(•, •) is T-coercive. What is less clear is the discrete counterpart of this approach. In this paper, we consider problem [START_REF] Ramakrishna | Physics of negative refractive index materials[END_REF] with the following hypothesis on σ:

σ |Ω1 = σ 1 is a constant such that σ 1 > 0, σ |Ω2 = σ 2 is a constant such that σ 2 < 0.
The ratio κ σ := σ 2 /σ 1 is called the contrast. Let Ω be a domain of R 2 , that is a connected bounded open subset of R 2 with a Lipschitz boundary. It is split as Ω = Ω 1 ∪ Ω 2 , where Ω 1 and Ω 2 are two disjoint domains.

The interface separating Ω 1 and Ω 2 is called Σ: we assume that it is a polygonal line made of straight edges and corners. Given v defined over Ω, we use the notation v i := v |Ωi , i = 1, 2. The equivalent variational formulation of (1) reads: Find u ∈ V such that ∀w ∈ V, a(u, w) = f, w ,

where V = H 1 0 (Ω), a is the form defined in [START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF], and •, • denotes the duality pairing between V and its dual V . Let us first describe a simple configuration for which everything is well understood. It is the symmetric geometry: Σ is a part of a straight line and Ω 2 is the symmetric of Ω 1 with respect to Σ. Then one can easily prove that the problem is well-posed if and only if κ σ = -1, by considering the following operators T:

T v = v 1 in Ω 1 -v 2 + 2Sv 1 in Ω 2 , respectively T v = v 1 -2Sv 2 in Ω 1 -v 2 in Ω 2 ,
where Sv 1 (x) = v 1 (Sx) (resp. Sv 2 (x) = v 2 (Sx)), with S denoting the symmetry with respect to Σ. On one hand, one can prove that a(•,

•) is T-coercive if κ σ = -1.
Then problem [START_REF] Bonnet-Ben Dhia | On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients[END_REF], which is equivalent to the problem Find u ∈ V such that ∀w ∈ V, a(u, T w) = f, T w , is well-posed. On the other hand, if κ σ = -1, the problem (1) is ill-posed since it has a kernel (set of solutions with zero right-hand side f ) which is infinite dimensional. Thus we say that the condition κ σ = -1 is optimal for the well-posedness of the continuous problem. Suppose now that κ σ = -1 and that we want to approximate the solution with a conforming finite element method (for short, a cFE method). This leads to the discrete problems

Find u h ∈ V h such that ∀w h ∈ V h , a(u h , w h ) = f, w h , (4)
where (V h ) h denotes a sequence of finite-dimensional subspaces of V , with h a positive parameter that goes to 0. If T(V h ) ⊂ V h for small h, then T-coercivity can be exploited at the discrete level. In particular, the discrete problem is well-posed and, by a straightforward adaptation of Céa's lemma, the error is controlled by the best approximation error. Hence, we just have to ensure the condition T(V h ) ⊂ V h , which is achieved in this symmetric geometry by using a symmetric mesh (for a uniform degree of approximation). Let us emphasize that using non-symmetric meshes can deteriorate drastically the convergence of the cFE method when κ σ is close to -1 (cf. [START_REF] Chesnel | T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients[END_REF]).

The objective of our paper is to generalize this type of result to an arbitrary polygonal interface. This is based on two ingredients:

1) adapting the condition T(V h ) ⊂ V h to elementary geometries where Σ includes a single corner ;

2) for every h, relaxing the condition T(V h ) ⊂ V h by defining a discrete operator

T h ∈ L(V h ) such that lim h→0 T h -T L(V h ;V ) = 0.
Finally, our aim is to provide meshing rules for an arbitrary geometry, ensuring that the standard convergence rate is recovered, as soon as κ σ is such that the continuous problem (1) is well-posed. Let us emphasize that in practice, the discrete problem ( 4) is implemented as it is, the operators (T h ) h being used only to prove convergence theoretically.

The outline of the paper is as follows. In the next section, we provide a review of the techniques proposed so far to approximate problem [START_REF] Ramakrishna | Physics of negative refractive index materials[END_REF] with sign-changing coefficients. Section 3 is dedicated to the construction of new explicit T-coercivity operators for elementary geometries whose interface has only one corner.

Then in section 4 we develop the theory that allows one to study the well-posedness of problem (1) for an arbitrary polygonal interface, and as a result we derive the optimal condition on the contrast κ σ to ensure well-posedness. Our aim is to provide tools that can be extended to the discrete problems, namely the approximation of problem (1) by cFE methods. This is the main topic of section 5, where convergence is derived as soon as the optimal condition on the contrast is fulfilled. Numerical results are presented in section 6.

Finally some concluding remarks are given.

Historical background: continuous and discrete T-coercivity

Let us review known results concerning the T-coercivity approach. Introduced in [START_REF] Bonnet-Ben Dhia | Time harmonic wave diffraction problems in materials with sign-shifting coefficients[END_REF], the T-coercivity technique has then been developed in [START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF]. Similarly to the symmetric case, the idea is to build linear continuous operators T on V :

T v = v 1 in Ω 1 -v 2 + 2Rv 1 in Ω 2 , respectively T v = v 1 -2R v 2 in Ω 1 -v 2 in Ω 2 , (5)
where R, R are operators (to be precised) such that Rv

1 | Σ = v 1 | Σ , R v 2 | Σ = v 2 | Σ .
This construction is attractive because once the operators R, R are settled, the operators T, T define isomorphisms of V . Let us present the available constructions of R (note that R is obtained by inverting the roles of Ω 1 and Ω 2 ).

The first idea (see [START_REF] Bonnet-Ben Dhia | Time harmonic wave diffraction problems in materials with sign-shifting coefficients[END_REF]) was to consider an operator R (0) that acts from H 1/2 00 (Σ) (the space of traces on Σ of functions of V ) to V 2 , where

V i := {v i , v ∈ V }, i = 1, 2.
One can choose for instance a harmonic lifting. Then it was proven that T-coercivity is realized with T (0) (defined as [START_REF] Bonnet-Ben Dhia | Time harmonic wave diffraction problems in materials with sign-shifting coefficients[END_REF] with R (0) ) under the condition that |κ σ | is "small enough" (with no explicit bound). At the discrete level, for

v h ∈ V h , R (0) h v h 1 was defined as the discrete-harmonic element of V h 2 (V h i := {v h i , v h ∈ V h }, i = 1, 2), with a trace on Σ equal to v h 1 |Σ .
A generalization was next proposed in [START_REF] Nicaise | A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients[END_REF][START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF] with an operator R that acts from V 1 to V 2 . Let us point out the difference between the two approaches: in T (0) , R (0) acts on the trace of v 1 on Σ, while in T (defined as [START_REF] Bonnet-Ben Dhia | Time harmonic wave diffraction problems in materials with sign-shifting coefficients[END_REF] with R), R acts on the whole function v 1 , defined on Ω 1 . The case of the symmetry-based operator S (see the introduction) fits into this second category but not in the first one. In that case, T-coercivity is realized under the condition

κ σ ∈ [-R 2 ; -1/ R 2 ], (6)
with( 1)

R = sup w1∈V1\{0} ∇Rw 1 Ω2 ∇w 1 Ω1
, and R = sup

w2∈V2\{0} ∇R w 2 Ω1 ∇w 2 Ω2 ; (7)
The authors of [START_REF] Nicaise | A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients[END_REF] exhibit a special geometry, different from the symmetric one, where the operators R, R can be again built using only axial symmetries. This is the case where Ω is a square, and Ω 1 is one quadrant of this square (right-angle geometry, see figure 1-left). One finds R2 = R 2 = 3. On the other hand, the case where Ω is a disk and Ω 1 is an angular sector of angle α is solved in [START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF] (see figure 1-right). For this second configuration the operators are based on the composition of a central symmetry and an angular dilation. In this case, R 2 = R 2 = I α where

Ω 1 Ω 1 Ω 2 Ω 2 α
I α := max 2π -α α , α 2π -α . ( 8
)
The result is optimal in the sense that the problem (1)-( 3) is ill-posed when κ σ ∈ [-I α ; -1/I α ], cf. [START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF]. Note that when α = π/2, one has I α = 3, and we remark that the interval in ( 6) is identical for both geometries.

The case of an arbitrary geometry was completely clarified in [START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF]. It appears that proving the wellposedness of (1)-( 3) is too restrictive, and possible only for very simple cases. A more relevant objective is to ensure that the problem is well-posed in the Fredholm sense (see section 4). And it has been proved by a localization process that this property depends only on the value of the contrast κ σ and on the geometry of the interface Σ, but not on the global geometry of Ω. The localization is realized with the help of suitably chosen cut-off functions which vanish away from the interface. In particular for a given closed polygonal line Σ, problem (3) is well-posed in the Fredholm sense if and only if κ σ ∈ [-I α ; -1/I α ] where α is the smallest corner angle of Σ and I α is defined as in [START_REF] Carvalho | Étude mathématique et numérique de structures plasmoniques avec des coins[END_REF].

What are the discrete counterparts of these results? The difficulty is that, in general, the imbedding T(V h ) ⊂ V h does not hold, due to the presence of the operators R, R , or due to the presence of the cut-off functions. To address this issue, various approaches have been proposed in [START_REF] Bonnet-Ben Dhia | Time harmonic wave diffraction problems in materials with sign-shifting coefficients[END_REF], [START_REF] Nicaise | A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients[END_REF] and [START_REF] Chesnel | T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients[END_REF]. We refer to the latter for the more complete and detailed numerical analysis. Roughly speaking, in all these previous works, the numerical analysis hinges on the introduction of an approximated operator T h , defined for instance with the help of an interpolation operator. In [START_REF] Bonnet-Ben Dhia | Time harmonic wave diffraction problems in materials with sign-shifting coefficients[END_REF] and [START_REF] Nicaise | A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients[END_REF], the drawback is that convergence of the cFE method is guaranteed under a condition on the contrast which is more restrictive than the one for the continuous problem. An attempt(2 ) to remove this limitation has been proposed in [START_REF] Chesnel | T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients[END_REF], but only for interfaces without corners. In the following we provide results for an arbitrary polygonal interface. We start with the case where the interface has only one corner.

New T-coercivity operators and associated T-conforming meshes around corners

Going back to the case of the disk (see figure 1-right), if α = π/2 there are at least two alternatives to build the operators T, T to prove well-posedness when κ σ ∈ [-3; -1/3]. The first one relies on axial symmetries (similarly to [START_REF] Nicaise | A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients[END_REF]), whereas the second one relies on a composition of a symmetry with respect to the center and an angular dilation [START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF]. Although these two approaches are equivalent at the continuous level, this is not the case at the discrete level. Indeed, for the first approach it is clear that T(V h ) ⊂ V h and T (V h ) ⊂ V h if one uses axially symmetric meshes. On the other hand, for the second approach T(V h ) ⊂ V h is never satisfied since the polynomial nature of basis FE functions is not preserved by angular dilations. The object of this section is to generalize the first approach to an arbitrary angle. To do so, first we build new operators R new , R new as rotation-and symmetry-based operators such that T-coercivity is realized (using T, T defined as [START_REF] Bonnet-Ben Dhia | Time harmonic wave diffraction problems in materials with sign-shifting coefficients[END_REF] with R new , R new ) under the same conditions as the case of an angular sector [START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF]. The key idea is to consider a pattern-based domain Ω (see figure 2). To define such a domain Ω, let p, q ∈ N \ {0} with p + q even and introduce the angle β := 2π p+q , and the cone C β := {(ρ cos θ, ρ sin θ) | 0 < ρ, 0 < θ < β}. Finally let us introduce the axial symmetries S j with respect to the line θ = jβ for j = 0, p + q -1 (the numbering is chosen counterclockwise). Let P ⊂ C β be a bounded domain that coincides locally at the corner with C β . A pattern-based domain Ω is the union of p + q patterns, i.e. Ω = p+q j=1 P j , where P 1 = P and P j+1 = S j P j for j = 1, p + q -1. The condition p + q even ensures that P 1 = S 0 P p+q . Then Ω 1 is composed of the first p patterns, Ω 1 = p j=1 P j , and Ω 2 is composed of the last q patterns, Ω 2 = p+q j=p+1 P j . In figure 2, one has p = 2 and q = 4 with respectively a sector, a triangle, a leaf as pattern P. Then one has α := pβ = 2π p p+q and it follows that

Σ α c Σ c α Σ α c
I α := max p q , q p . (9)
Note that for the disk, the condition on the parity of p + q is by no means restrictive. Indeed if p + q is odd, one simply doubles p and q without changing the value of α.

For the sake of clarity, we introduce also the following notations:

• Ω k 1 = P k , k = 1, p, the k-th pattern of Ω 1 .
• Ω l 2 = P p+l , l = 1, q, the l-th pattern of Ω 2 .

• v k 1 := v| Ω k 1 , k = 1, p the restriction to Ω k 1 . Similarly, v l 2 := v| Ω l 2 , l = 1, q.
• e n , n = 1, 2, the two edges of Σ such that e 1 coincides locally with {(ρ, 0), 0 < ρ}, while e 2 coincides locally with the line {(ρ cos α, ρ sin α), 0 < ρ}.

To define R new as a rotation-and symmetry-based operator from V 1 to V 2 , one introduces R m the rotation of angle mβ, m = 0, p + q -1. Define also their inverse R -m the rotation of angle -mβ. These geometrical transforms are such that

• for (ρ, θ) ∈ Ω k 1 , for all k ∈ I 1 := 1, min(p, q) , S 0 (ρ, θ) ∈ Ω q+1-k 2 , • for (ρ, θ) ∈ Ω k 1 , for all k ∈ I 2 := p + 1 -min(p, q), p , S p (ρ, θ) ∈ Ω p+1-k 2 , • for (ρ, θ) ∈ Ω k 1 , for all k ∈ 1, p , R p+l-k (ρ, θ) ∈ Ω l 2 , for all l ∈ 1, q .
Then one defines symmetry-based operators S n (n = 1, 2) and rotation-based operators R m (m = 1, p + q -1)

from V k 1 := {v k 1 , v ∈ V } to V l 2 := {v l 2 , v ∈ V } by: • S 1 v k 1 (ρ, θ) = v k 1 (S 0 (ρ, θ)) for (ρ, θ) ∈ Ω q+1-k 2 , for all k ∈ I 1 , • S 2 v k 1 (ρ, θ) = v k 1 (S p (ρ, θ)) for (ρ, θ) ∈ Ω p+1-k 2 , for all k ∈ I 2 , • R k-(p+l) v k 1 (ρ, θ) = v k 1 (R k-(p+l) (ρ, θ)) for (ρ, θ) ∈ Ω l 2
, for all l = 1, q, for all k = 1, p.

A rotation-and symmetry-based operator is then an ad hoc composition of S 1 , S 2 and R m . The general construction of a global, pattern-based, admissible operator

R from V 1 to V 2 , ensuring Rv |Σ = v |Σ is given in the Appendix A.
1. We give one illustrative example below with p < q, which is equivalent to α < π: we build admissible operators R and R in this case. We emphasize that the construction of an operator R in the case α < π can be used to build an operator R in the case α > π, and vice versa, cf. [START_REF] Bonnet-Ben Dhia | Time harmonic wave diffraction problems in materials with sign-shifting coefficients[END_REF]. We consider α = 2π/3, with p = 2 and q = 4. Construction of operators R: one can define two admissible operators (illustrated in figure 3):

R adm 1 v 1 =          S 2 v 2 1 in Ω 1 2 R -2 v 2 1 in Ω 2 2 S 1 v 2 1 in Ω 3 2 S 1 v 1 1 in Ω 4 2
, and R adm 2 To realize T-coercivity under the same conditions as in [START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF], one has to check that these operators R are optimal in the sense that R 2 = I α , where I α is defined in ( 8) or ( 9), while the operator norm is defined in [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF]. For the present example, I α = 2. Since S n and R -m (n = 1, 2, m = 1, 5) are isometry-based operators, one finds easily that

v 1 =          S 2 v 2 1 in Ω 1 2 S 2 v 1 1 in Ω 2 2 R -4 v 1 1 in Ω 3 2 S 1 v 1 1 in Ω 4 2 . Σ c v 1 1 v 2 1 S 1 v 1 1 S 1 v 2 1 R -2 v 2 1 S 2 v 2 1 Σ c v 1 1 v 2 1 S 1 v 1 1 R -4 v 1 1 S 2 v 1 1 S 2 v 2 1 e 1 e 1 e 2 e 2
∇(R adm 1 v 1 ) 2 Ω2 = 3 ∇v 2 1 2 Ω 2 1 + ∇v 1 1 2 Ω 1 1 ≤ 3 ∇v 1 2 Ω1 , and
∇(R adm 2 v 1 ) 2 Ω2 = ∇v 2 1 2 Ω 2 1 +3 ∇v 1 1 2 Ω 1 1 ≤ 3 ∇v 1 2 Ω1 ,
with equality for appropriately chosen v 1 , for instance v 1 such that v 2 1 = 0 when evaluating R adm

2

. According to ( 6)-( 7), this gives R adm 1 2 = R adm 2 2 = 3, which is greater than the expected optimal value of the norm squared. However, considering the average of these two admissible operators gives us the result. More precisely, define

R new v 1 := 1 2 (R adm 1 + R adm 2 )v 1 =                S 2 v 2 1 in Ω 1 2 1 2 (R -2 v 2 1 + S 2 v 1 1 ) in Ω 2 2 1 2 (S 1 v 2 1 + R -4 v 1 1 ) in Ω 3 2 S 1 v 1 1 in Ω 4 2 , then one can prove that R new 2 = 2. Indeed, for v 1 ∈ V 1 ∇(R new v 1 ) 2 Ω2 = ∇(S 2 v 2 1 ) 2 Ω 1 2 + 1 2 ∇(R -2 v 2 1 + S 2 v 1 1 ) 2 Ω 2 2 + 1 2 ∇(S 1 v 2 1 + R -4 v 1 1 ) 2 Ω 3 2 + ∇(S 1 v 1 1 ) 2 Ω 4 2
and since S n and R -m (n = 1, 2, m = 1, 5) are isometry-based operators, using the triangle inequality one finds that

∇(R new v 1 ) 2 Ω2 ≤ ∇v 2 1 2 Ω 2 1 + ( 1 2 ∇v 2 1 Ω 2 1 + 1 2 ∇v 1 1 Ω 1 1 ) 2 + ( 1 2 ∇v 2 1 Ω 2 1 + 1 2 ∇v 1 1 Ω 1 1 ) 2 + ∇v 1 1 2 Ω 1 1 . Define the matrix M =     0 1 1/2 1/2 1/2 1/2 1 0     , and
-→ W ∈ R 2 such that -→ W := ( ∇v 1 1 Ω 1 1 , ∇v 2 1 Ω 2 1 ) ; by construction, -→ W 2 = ∇v 1 Ω1 , where • 2 denotes the Euclidean norm. Then one has M -→ W = ( ∇v 2 1 Ω 2 1 , 1 2 ∇v 1 1 Ω 1 1 + 1 2 ∇v 2 1 Ω 2 1 , 1 2 ∇v 1 1 Ω 1 1 + 1 2 ∇v 2 1 Ω 2 1 , ∇v 1 1 Ω 1 1
) and one remarks that

∇(R new v 1 ) 2 Ω2 ≤ M -→ W 2 2 = (M M -→ W , -→ W ) ≤ M M 2 ∇v 1 2 Ω1 . Hence R new 2 ≤ M M 2 . A straightforward computation shows that M M 2 = 2.
Finally, the equality R new 2 = M M 2 can be recovered easily( 3 ).

3 For instance, equality is obtained for Rnew by choosing

v 1 such that v k 1 ∈ H 1 0 (Ω k 1 ), for k = 1, p, with v 1 1 a symmetric function w.r.t. the line θ = β 2 , and v k 1 (ρ, θ) = v k 1 (R k-1 (ρ, θ)), for (ρ, θ) ∈ Ω k 1 , for k = 2, p. Similarly, equality for R new is obtained by choosing v 2 such that v l 2 ∈ H 1 0 (Ω l 2 ), for l = 1, q, with v 1 2 a symmetric function w.r.t. the line θ = (p + 1 2 )β, and v l 2 (ρ, θ) = v 1 2 (R l-1 (ρ, θ)), for (ρ, θ) ∈ Ω l 2 , for l = 2, q.
Construction of operators R : in that case one "folds v 2 up" such that R v 2 is defined over Ω 1 . Then one obtains the two admissible operators described in figure 4: One can operate similarly for the second example by considering R new v 2 over Ω 1 as the average of the two operators described in figure 4:

R adm 1 v 2 = S 1 v 4 2 -R 2 v 3 2 + S 2 v 2 e 1 e 2
R new v 2 =      1 2 (2S 1 v 4 2 -R 2 v 3 2 + S 2 v 2 2 ) in Ω 1 1 1 2 (2S 2 v 1 2 -R 4 v 2 2 + S 1 v 3 2 ) in Ω 2 1 . Then one finds R new 2 ≤ M M 2 = 2, with the matrix M = 0 1/2 1/2 1 1 1/2 1/2 0 = M . Remark that
it is expected to find that M = M as the role of p and q are exchanged. Thus, one obtains the same results for R new . We note that equality for the norms, that is R new 2 = M M 2 , can be recovered easily( 3 ). Hence, R new is an optimal operator. Summing up, for any p, q, one simply defines R new , R new as the average of all admissible rotation-and symmetry-based operators. The general expression of these operators is given in the Appendix A.1 and propositions 2 and 3 there give us that for any p, q, max(

R new 2 , R new 2 ) = I α .
Let us make some comments regarding this approach:

-this approach applies when the corner angle α can be expressed as a rational fraction times 2π, that is α ∈ 2πQ. In the case of an irrational angle α, since Q is dense in R, one can come as close to α as desired (with an increase of the value of p + q).

-in the case of a general polygonal interface Σ, Ω is locally pattern-based in a neighborhood of any interior corner. Consequently this approach can be adapted using a localization process (see next section).

At this point we have provided optimal operators, that is operators with norm-squared equal to I α . Finally we explain how it is possible to satisfy T(V h ) ⊂ V h and T (V h ) ⊂ V h , with T, T defined as (5) with these optimal operators. Suppose that Ω is a pattern-based domain, and that the pattern is polygonal (for instance a triangle). Once this pattern is meshed, the requirement to get T-conforming meshes is to duplicate by symmetry this mesh in each pattern successively. When one considers a uniform degree of approximation on the whole domain, one has automatically T(V h ) ⊂ V h and T (V h ) ⊂ V h with the new operators described above. Then, as already mentioned, classical errors estimates directly follow from Céa's lemma [11, Corollary 1]. Note that there is no need for additional symmetry requirements for the meshing of the pattern.

Numerical illustrations.

We consider the geometry described previously: Ω hex is a hexagon where Ω hex 1 locally coincides with a cone of angle α = 4π/3. In this case I α = 2 and [-I α ; -1/I α ] = [-2; -1/2]. Let us construct an exact solution (of a problem of type (1) with κ σ ∈ [-2; -1/2], see [START_REF] Chesnel | Étude de quelques problèmes de transmission avec changement de signe[END_REF]

below): consider u r ∈ H 1 (Ω hex ) such that in polar coordinates( 4 ) u r (ρ, θ) = σ -1 1 ρ 2 sin( p+q 2 (θ-α)) in Ω hex 1 , σ -1 2 ρ 2 sin( p+q 2 (θ-α)) in Ω hex 2 ,
where p (resp. q) still denotes the number of patterns in

Ω hex 1 (resp. Ω hex 2 ) and f := -div(σ∇u r ) = 1 4 ((p + q) 2 -16) sin( p+q 2 (θ-α)) ∈ L 2 (Ω hex ). By construction, u r is piecewise smooth [17]: u r | Ω hex i ∈ H 3-ε (Ω hex i ), ∀ε > 0, i = 1, 2.
To illustrate the importance of T-conforming meshes around corners, let us add to u r a singular part, that is some u s (ρ, θ) = ρ λ Φ(θ), with λ := λ(σ) ∈ R, such that div(σ∇u s ) = 0 in Ω hex . For example we consider [START_REF] Bonnet-Ben Dhia | Analyse spectrale et singularités d'un problème de transmission non coercif[END_REF][START_REF] Nicaise | A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients[END_REF]

Φ(θ) =            cosh(λ(θ -α/2)) cosh(λα/2) 0 ≤ θ ≤ α/2 cosh(λ(θ + (2π -α)/2)) cosh(λ(2π -α)/2) -(2π -α)/2 ≤ θ ≤ 0 , Φ(θ) = Φ(2π -θ -α/2) -π ≤ θ ≤ -(2π -α) or α/2 ≤ θ ≤ π.
One chooses for λ the smallest possible positive real number. Its value depends on κ σ , and one can prove that since

κ σ ∈ [-2; -1/2], u s ∈ H 1 (Ω hex ) [3], with u s | Ω hex i ∈ H 1+λ-ε (Ω hex i ), ∀ε > 0, i = 1, 2. Defining g := (u r + u s )| ∂Ω hex , one checks that u = u r + u s is the unique solution of the problem      Find u ∈ H 1 (Ω hex ) such that: -div(σ∇u) = f in Ω hex u = g on ∂Ω hex . ( 10
)
With the help of a lifting of the non-zero boundary condition, one easily checks that problem [START_REF] Chesnel | Étude de quelques problèmes de transmission avec changement de signe[END_REF] set in this hexagonal domain is well-posed when κ σ ∈ [-2; -1/2]. We consider two kinds of meshes (see figure 5), namely standard meshes (without rotation-or symmetry-based invariance) and T-conforming meshes, and discretize the problem for several FE orders, for two chosen contrasts: κ σ = -3 and κ σ = -2.1 (see figure 6). For this geometry, computations give λ = 0.5 when κ σ = -3, and λ 0.205 when κ σ = -2.1. Classically the regularity of the solution u = u r + u s is driven by the singular behavior u s . Consequently one expects an order of convergence equal to λ for the relative errors in H 1 -norm. With figure 6 one concludes that T-conforming meshes ensure optimal convergence speed while, for standard meshes, convergence is more erratic with respect to the mesh size. Note that in figure 6 one does not improve the order of convergence using higher FE orders, due to the low regularity of the solution. In a more general case of a polygonal interface with several interior corners, one has to apply locally this tilings method in the neighborhood of the corners. This strategy is explained in the next two sections.

Weak T-coercivity for a general polygonal interface

In this section we recall some theoretical results regarding the T-coercivity approach, and prove well-posedness of problem (1) for an arbitrary geometry using the new geometry-based operators introduced in section 3. Consider a Hilbert space E with its dual E , a bilinear form b defined over E × E and B a (linear continuous operator) from E to E such that Bv, w = b(v, w), for all v, w ∈ E. Then, for some data f ∈ E , solving

Find u ∈ E such that b(u, w) = f, w , ∀w ∈ E, (11)
is equivalent to solving

Find u ∈ E such that Bu = f in E'. (12)
Classically [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF][START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF], we recall that B is said to be a Fredholm operator when dim(ker(B)) < ∞, its range R(B) is closed and codim(R(B)) < ∞ ; in this case its index is equal to dim(ker(B)) -codim(R(B)). If in addition the associated form b is hermitian, the index is automatically equal to 0. When B is a Fredholm operator of index 0, we say that ( 11)-( 12) is well-posed in the Fredholm sense. On the other hand, ( 11)-( 12) is well-posed if, and only if, for all f ∈ E , it has one and only one solution u, with continuous dependence: there exists C > 0 such that, for all f ∈ E , the solution u verifies u E ≤ C f E . In terms of operators, it means that B -1 is well-defined as a continuous operator from E to E.

To prove well-posedness (in the Fredholm sense) of problem (1), we will apply the theory of T-coercivity [START_REF] Bonnet-Ben Dhia | Time harmonic wave diffraction problems in materials with sign-shifting coefficients[END_REF][START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF][START_REF] Chesnel | T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients[END_REF]. Let us recall some results. Within the Banach-Necas-Babuska framework, one can define a weak stability condition, also called an inf-sup condition. Below, L(E) is the set of linear continuous operators defined on E, and K(E) is the subset of compact operators.

∃C ∈ K(E), ∃α > 0, β ∈ R, ∀v ∈ E, sup w∈E\{0} |b(v, w)| w E ≥ α v E -β Cv E . (13)
Let us now introduce an a priori intermediate condition (cf. [START_REF] Bonnet-Ben Dhia | Time harmonic wave diffraction problems in materials with sign-shifting coefficients[END_REF]).

Definition 2. Let b(•, •) be a continuous sesquilinear form on E × E. It is weakly T-coercive if ∃C ∈ K(E), ∃T ∈ L(E) bijective, ∃α > 0, β ∈ R, ∀v ∈ E, |b(v, Tv)| ≥ α v 2 E -β Cv 2 E . (14)
In other words, the form b(•, T•) fulfills a Gärding's inequality [START_REF] Schatz | An observation concerning Ritz-Galerkin methods with indefinite bilinear forms[END_REF]. As seen in the introduction, problem (1) set in the Hilbert space V = H 1 0 (Ω) can be expressed in variational form as [START_REF] Bonnet-Ben Dhia | On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients[END_REF]. Let us introduce the linear continuous operator A from V to V such that Av, w = a(v, w) for all v, w ∈ V , with the form a defined in (2): by construction, A : v → -div(σ∇v). In operator form, (1)-(3) writes equivalently Find u ∈ V such that Au = f in V .

We propose below some realizations of the inf-sup operator that can be used for problem (1) for a polygonal interface Σ. From now on, we suppose for simplicity that Σ is a polygonal line without endpoints, ie. it is a loop. Let N denote the number of its corners, (c n ) n=1,N its corners, (α n ) n=1,N the corner angles measured in Ω 1 , and (e n ) n=1,N its edges. We introduce the polar coordinates (ρ n , θ n ) centered at c n such that Ω 1 coincides locally with the cone C αn . First, we build a partition of unity on Ω. So, let

(χ n ) n=1,2N ∈ (C ∞ (Ω; [0, 1]
)) N with supports localized in a neighborhood of the interface, such that, for n = 1, N , χ n = 1 near the corner c n and χ n = 0 "far" from c n (in particular, (χ n ) n=1,N have mutually disjoint supports); respectively χ n+N = 0 on Σ \ e n . Plus, one imposes the condition n=1,2N χ n = 1 on Σ. Then we define χ 0 = 1 -n=1,2N χ n , which vanishes in a neighborhood of the interface Σ and, setting P = 2N , we obtain that (χ p ) p=0,P ∈ (C ∞ (Ω; [0, 1])) P +1 is a partition of unity on Ω. Finally we denote by S p := supp(χ p ) the support of the cut-off function χ p . Now, let (R p ) p=1,P be linear continuous operators that act from

V 1 to V 2 (resp. (R p ) p=1,P from V 2 to V 1 ).
We suppose in addition that these operators are such that, for all p = 1, P ,

∃C, C > 0, ∀w 1 ∈ V 1 , R p w 1 Ω2 ≤ C w 1 Ω1 and χ 1/2 p ∇(R p w 1 ) Ω2∩Sp ≤ C χ 1/2 p ∇w 1 Ω1∩Sp , ∃C, C > 0, ∀w 2 ∈ V 2 , R p w 2 Ω1 ≤ C w 2 Ω2 and χ 1/2 p ∇(R p w 2 ) Ω1∩Sp ≤ C χ 1/2 p ∇w 2 Ω2∩Sp . (16)
We introduce

R := max p=1,P R p , R := max p=1,P R p , resp. |R| := max p=1,P |R p |, |R | := max p=1,P |R p |, (17)
where for all p = 1, P ,

R p := sup w1∈V1, χ 1/2 p ∇w1 Ω 1 ∩Sp =1 χ 1/2 p ∇R p w 1 Ω2∩Sp , |R p | := sup w1∈L 2 (Ω1), w1 Ω 1 =1 R p w 1 Ω2 , R p := sup w2∈V2, χ 1/2 p ∇w2 Ω 2 ∩Sp =1 χ 1/2 p ∇R p w 2 Ω1∩Sp , |R p | := sup w2∈L 2 (Ω2), w2 Ω 2 =1 R p w 2 Ω1 . (18)
Finally, we assume matching conditions on the traces:

∀p, ∀v 1 ∈ V 1 , R p v 1 | Σ∩Sp = v 1 | Σ∩Sp , ∀p, ∀v 2 ∈ V 2 , R p v 2 | Σ∩Sp = v 2 | Σ∩Sp . ( 19
)
Remark 2. At the end of the section, we will show how in practice we can provide χ p , R p , R p which fulfill all the conditions above.

Finally we define the two operators

Tv =    v 1 on Ω 1 -v 2 + 2 p=1,P χ p R p v 1 on Ω 2 , T v =    v 1 -2 p=1,P χ p R p v 2 on Ω 1 -v 2 on Ω 2 . ( 20
)
Now one can prove the following Lemma 2. Suppose that χ p , R p , R p satisfy (16) for all p = 1, P . If the contrast κ σ does not belong to [-R 2 ; -1/ R 2 ] where R and R are defined by ( 17)-( 18), then the form a is weakly T-coercive for T or T defined in [START_REF] Ramdani | Lignes supraconductrices : analyse mathématique et numérique[END_REF], and problem (1) is well-posed in the Fredholm sense.

Remark 3. According to [START_REF] Costabel | A direct boundary integral method for transmission problems[END_REF][START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF], the case κ σ = -1 leads to an ill-posed problem (1) in any geometry. Consequently, it follows that R ≥ 1, R ≥ 1.

Proof. We assume for instance that κ σ ∈ ( -1/ R 2 ; 0). By lemma 1, we just have to show that the form a is weakly T-coercive, namely

∃C ∈ K(V ), ∃T ∈ L(V ), ∃α > 0, β ∈ R, ∀v ∈ V, a(v, Tv) ≥ α v 2 V -β Cv 2 V .
We consider the operator T defined in (20)-left to prove the above condition. Due to the matching conditions [START_REF] Nicaise | A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients[END_REF] satisfied by (R p ) p , Tv ∈ V for all v ∈ V and, in addition one checks easily that T • T = I V so T is bijective. Then: a(v, Tw) = b(v, w) + c(v, w), where the forms b and c are respectively defined by:

b(v, w) = |σ 1 |(∇v 1 , ∇w 1 ) Ω1 + |σ 2 |(∇v 2 , ∇w 2 ) Ω2 -2|σ 2 | p=1,P (∇v 2 , χ p ∇(R p w 1 )) Ω2∩Sp , c(v, w) = -2|σ 2 | p=1,P (∇v 2 , R p w 1 ∇χ p ) Ω2∩Sp . First we prove that b is coercive. Using Young's inequality with η > 0 on b(v, v), we get b(v, v) ≥ |σ 1 | ∇v 1 2 Ω1 + |σ 2 | ∇v 2 2 Ω2 -|σ 2 | p=1,P η χ 1/2 p ∇v 2 2 Ω2∩Sp + η -1 χ 1/2 p ∇(R p v 1 ) 2 Ω2∩Sp .
One has:

p=1,P χ 1/2 p ∇v 2 2 Ω2∩Sp ≤ p=1,P χ 1/2 p ∇v 2 2 Ω2 = p=1,P (χ p ∇v 2 , ∇v 2 ) Ω2 = ((1 -χ 0 )∇v 2 , ∇v 2 ) Ω2 ≤ ∇v 2 2 Ω2 , as 1 -χ 0 L ∞ (Ω2) = 1.
Next, using the definitions of ( R p ) p=1,P and R , we note that:

p=1,P χ 1/2 p ∇(R p v 1 ) 2 Ω2∩Sp ≤ p=1,P R p 2 χ 1/2 p ∇v 1 2 Ω1∩Sp ≤ R 2 p=1,P χ 1/2 p ∇v 1 2 Ω1 = R 2 ((1 -χ 0 )∇v 1 , ∇v 1 ) Ω1 ≤ R 2 ∇v 1 2 Ω1 . Hence, b(v, v) ≥ (|σ 1 | -|σ 2 | η -1 R 2 ) ∇v 1 2 Ω1 + |σ 2 |(1 -η) ∇v 2 2 Ω2 . Because κ σ ∈ ( -1/ R 2 ; 0), we can choose η such that R 2 |κ σ | < η < 1, which is equivalent to |σ 1 | -|σ 2 | η -1 R 2 > 0 and |σ 2 |(1 -η) > 0. Regarding c(v, v), if we let C : V → V such that (Cv, w) V = c(v, w) for all v, w ∈ V , then by Cauchy-Schwarz inequality one gets Cv 2 V = c(v, Cv) ≤ G ∇v Ω Cv Ω , where G := 2P |R| |σ 2 | max p=1,P (|χ p | W 1,∞ (Ω2) )
. By Rellich's theorem, one concludes that C is compact. Then by using Young's inequality with η > 0 we find

c(v, v) ≥ - 1 2 ((η ) -1 Cv 2 V + η v 2 V ),
which ends the proof in the case κ σ ∈ ( -1/ R 2 ; 0), by choosing η small enough.

On the other hand, if κ σ ∈ ( -∞; -R 2 ), one can reverse the roles of Ω 1 and Ω 2 by using this time the operator T defined in (20)-right. The proof then proceeds as above to prove that there exist α > 0, β , and a compact operator C such that

∀v ∈ V, a(v, T v) ≥ α v 2 V -β C v 2
V , which is the condition [START_REF] Ciarlet | Robust a posteriori error control for transmission problems with sign-changing coefficients using localization of dual norms[END_REF].

To conclude the study of the well-posedness of problem (1), let us explicit χ p , (R p ) p=1,P , (resp. (R p ) p=1,P ), and compute the bounds. In the case where Σ is a polygonal line with all angles α n ∈ 2πQ, n = 1, N , one can explicit these bounds using the results of section 3. To do so, let us define (B p ) p=1,P a sequence of connected open sets so that p=1,P B p is a neighborhood of the interface. For all n = 1, N , define B n as a triangle-based neighborhood of c n , and B N +n as a neighborhood of a part of e n (excluding its endpoints) which is symmetric with respect to e n (see figure 7). More precisely, for all n = 1, N , B n is a cyclic polygon centered at c n composed of p n > 0 triangles in Ω 1 and q n > 0 triangles in Ω 2 , with p n +q n even: define ρ cn the radius and s n the side length of this polygon. Then for all n = 1, N one defines B N +n as a trapezoid-based open set with one trapezoid in Ω 1 and one in Ω 2 , each trapezoid being of side lengths s n and s n+1 (5 ). For technical purposes one chooses p n ≥ 2 so that all part edge neighborhoods B N +n , n = 1, N are mutually disjoint open sets. Note that a part edge neighborhood intersects with two corners neighborhoods. Namely

for n = 1, N -1, one has B N +n ∩ B n = ∅, B N +n ∩ B n+1 = ∅, and for n = N one has B P ∩ B N = ∅, B P ∩ B 1 = ∅. Then one defines (R p ) p=1,P from V 1 to V 2 (and also from L 2 (Ω 1 ) to L 2 (Ω 2 )) such that for all w 1 ∈ V 1 and for all n = 1, N R n w 1 (x) =      0 if x ∈ B n R n new w 1 (x) if x ∈ B n , (21)
with R n new defined as in section 3 (the general expression is given in Appendix A.1), and

R N +n w 1 (x) =      0 if x ∈ B N +n w 1 (x Σ , -y Σ ) if x ∈ B N +n , (22)
where (x Σ , y Σ ) denotes the local cartesian coordinates, and the first axis coincides with Σ.

Finally, let us precise some properties the cut-off functions (χ p ) p . Recall that (ρ p , θ p ) denote the polar coordinates centered at c p , p = 1, N . For all p = 1, N , χ p (x) = χ p (ρ p ), and χ p = 1 for 0 ≤ ρ p ≤ ρ min,p for some ρ min,p > 0 ; respectively χ N +p is a symmetric function with respect to the interface. This can be realized by introducing ψ ∈ C ∞ (Ω; [0, 1]) equal to 1 in a neighborhood of the interface, for which we remark that ψ -p=1,N χ p is a (smooth) function that vanishes in a neighborhood of the corners, and also that "close" to the interface, it is a function which is symmetric with respect to the interface( 6). Then one chooses S p the support of χ p such that S p ⊂ B p for the corner case (p = 1, N )( 7) and in the edge case S p ⊂ B p ∪ B p-N ∪ B p+1-N (p = N + 1, P -1), resp. S P ⊂ B P ∪ B N ∪ B 1 (see figure 7). One can check that ( 19) is satisfied for all R p , p = 1, P . We can finally state the result on the well-posedness of problem [START_REF] Ramakrishna | Physics of negative refractive index materials[END_REF]. Proof. With the operators ( 21)-( 22) and using propositions 4 and 5 (cf. Appendix A.1), one finds that 1) is not well-posed in the Fredhom sense, according to section 6 of [START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF]. Hence, we conclude that the condition on the contrast κ σ ∈ [-I α ; -1/I α ] is optimal for a 2D geometry with a polygonal interface separating the two subdomains. We define then the critical interval

Ω 1 Ω 2 c 2 c 1 e N e 1 e 2 B 1 B 2 B N +1 B P B N +2
min( R 2 , R 2 ) ≥ 1 and max( R 2 , R 2 ) ≤ I α . Consequently [-R 2 ; -1/ R 2 ] ⊆ [-I α ; -1/I α ]. Then if κ σ ∈ [-I α ; -1/I α ] the result follows from lemma 2. Note that if κ σ ∈ [-I α ; -1/I α ], then the problem (
I c := [-I α ; -1/I α ].

T-conforming meshes for a general polygonal interface and error estimates

We study the approximation of problem (1) when the contrast κ σ does not belong to the critical interval [-I α ; -1/I α ]. According to the previous section (see theorem 1), it follows that problem (1) is well-posed in the Fredholm sense (namely there might be a finite dimensional kernel). From now on, we impose further that it is well-posed, to ensure the uniqueness of the solution u to be approximated. We study the family of discrete problems (4), focusing on conforming( 8) finite element approximations. Our aim is to prove that they can be solved in geometries with interfaces that include corners, and to prove optimality of the approximation, that is the unique solution u h tends towards the exact solution u when h tends to 0 with optimal convergence rate. The geometry and notations are those of section 4.

To fix ideas, we assume throughout section 5 that κ σ ∈ (-1/I α ; 0) so that we can use the operator T as defined in (20)-left. The conforming approximations in V are defined on meshes that match with the interface (all elements are subsets of either Ω 1 or Ω 2 ), with piecewise polynomial approximations. Let us consider (T h ) h a regular family of meshes of Ω, made of triangles ; for all triangles τ , one has either τ ⊂ Ω 1 or τ ⊂ Ω 2 . Define the family of Lagrange FE spaces (for a degree of approximation d ≥ 1)

V h (d) := v ∈ V : v |τ ∈ P d (τ ), ∀τ ∈ T h ,
where P d (τ ) is the space of polynomials of degree at most d on the triangle τ . Unless otherwise specified, the index (d) is omitted. The parameter, or meshsize, h is defined as h := max τ ∈T h h τ , where h τ is the diameter of τ . We recall that V h 1 (resp. V h 2 ) denotes the FE subspace of V 1 (resp. V 2 ) built on the triangulation T h . Throughout this section, C is used to denote a generic positive constant which is independent of the meshsize, the triangulation, and the data/unknown of interest. Definition 3. Given h, let T p h,i := {τ ∈ T h : τ ∩ int(S p ) ∩ Ω i = ∅}, for i = 1, 2 and p = 1, P . The meshes (T h ) h are locally T-conform if, for all h small enough, for all p = 1, P , for all τ ∈ T p h,1 , the image of τ by the geometrical transforms underlying R p belongs to T p h,2 . In practice, the above definition simply imposes that the meshes are locally: invariant by the axial symmetry for each part edge ; invariant by the axial symmetries S j from one pattern, P j , to the next, P j+1 , for j = 1, p + q -1, at each corner. Note that if the conditions of the above definition are met, then for all v h ∈ V h , it holds that R p v h 1 is equal to the restriction of some element of V h 2 on Ω 2 ∩ S p . We set

Ω h 2 := int( p=1,P ∪ τ ∈T p h,2 τ ).
As mentioned in section 2, in general, for v h ∈ V h , Tv h ∈ V h due to the cut-off functions (χ p ) p=1,P that need to be interpolated. In other words, the condition T(V h ) ⊂ V h is false in general. However one can look for a relaxed condition guaranteeing optimality of the approximation by introducing a discrete counterpart T h of T such that T h (V h ) ⊂ V h and lim h→0 T h -T L(V h ) = 0. Our goal in this section is to detail these steps, using some discrete operators (T h ) h (precised below).

In [START_REF] Chesnel | T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients[END_REF] it was proposed to replace χ p by its interpolation of degree 1 denoted by χ p,h . To fit with the definition of T in ( 20)-left, it leads to consider:

T 0 h v h = v h 1 in Ω 1 -v h 2 + 2 p=1,P χ p,h R p v h 1 in Ω 2 . ( 23
)
Unfortunately, for all p = 1, P , the degree of χ p,h R p v h 1 (restricted to a triangle) is too high. For instance, if one is dealing with Lagrange FE of degree 1 (d = 1), its degree is 2: hence, T 0 h v h does not belong to V h in general. This is why we introduce instead the discrete operator:

T h v h = v h 1 in Ω 1 -v h 2 + 2I h ( p=1,P χ p,h R p v h 1 ) in Ω 2 , ( 24
)
with I h the Lagrange-interpolation operator on V h (d) . Moreover from [16, Corollaries 1.109-1.110] we have for p = 1, P , the stability and approximation estimates

∃C > 0, χ p,h W 1,∞ (Ω2) ≤ C χ p W 1,∞ (Ω2) , ( 25
)
∃C > 0, ∀h small enough, χ p -χ p,h W 1,∞ (Ω2) ≤ C h |χ p | W 2,∞ (Ω2) . ( 26
)
We emphasize that the analysis presented in [START_REF] Chesnel | T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients[END_REF] remains valid if one replaces T 0 h by T h , as we explain below. Our aim is to prove that the form a is uniformly weakly T-coercive, namely

∃α > 0, β∈ R, ∀h small enough, ∀v h ∈ V h , |a(v h , T h v h )| ≥ α v h 2 V -β Cv h 2 V , (27)
and from there one can follow the last part of the proof of Proposition 3 in [START_REF] Chesnel | T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients[END_REF] to derive a uniform discrete inf-sup condition (by contradiction). Classically, this yields an error estimate.

In order to obtain (27), we shall evaluate |a(v h , (T -T h )v h )| for all v h ∈ V h . More precisely we want to prove

∃C 0 > 0, ∀h small enough, ∀v h ∈ V h , |a(v h , (T -T h )v h )| ≤ C 0 h v h 2 V , (28)
which is a consequence of proving that there exists C > 0 such that, for all h small enough, for all

v h ∈ V h , (T -T h )v h V ≤ C h v h V .
This result is the object of lemma 3.

Lemma 3. Assume that the meshes are locally T-conform then there holds

∃C > 0, ∀h small enough, ∀v h ∈ V h , (T -T h )v h V ≤ C h v h V , (29)
so the form a is uniformly weakly T-coercive.

Proof. Given v h ∈ V h , let us apply the triangle inequality:

(T -T h )v h V ≤ (T -T 0 h )v h V + (T 0 h -T h )v h V .
Using the continuity of (R p ) p=1,P and (26), one has for the first term of the right-hand side:

(T -T 0 h )v h V ≤ C p=1,P χ p -χ p,h W 1,∞ (Ω2) v h V ≤ C h v h V .
On the other hand, for the last term of the right-hand side setting w h p,2 = R p v h 1 , p = 1, P , one finds

(T 0 h -T h )v h V ≤ 2 p=1,P ∇(χ p,h w h p,2 -I h (χ p,h w h p,2 )) Ω h 2 .
For every p, since the meshes are locally T conform, the discrete function w h p,2 is equal to the restriction on Ω h 2 of an element of V h 2 . Using lemma 5 in §A.3 and the continuity of the operators (R p ) p=1,P , one finds

(T 0 h -T h )v h V ≤ C h v h V ,
which leads to (29), and thus to (28). It now follows that

∃α > 0, β∈ R, ∀h small enough , ∀v h ∈ V h , |a(v h , T h v h )| ≥ (α -C 0 h) v h 2 V -β Cv h 2 V ,
with α and β constants that express the weak T-coercivity of a(•, •) with the operator T as in (20)-left, see lemma 2 and theorem 1. Hence, one finds the desired result ( 27), which at the end yields the error estimate (cf. [START_REF] Chesnel | T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients[END_REF]).

Theorem 2. Assume that problem (1) is well-posed and that the meshes are locally T-conform. Then for h small enough there exists one and only one solution u h to (4), with the estimate

u -u h V ≤ C inf v h ∈V h u -v h V . (30)
We conclude that we have optimality of the approximation for problem (1), thanks to the choice of the discrete T-coercivity operator as in (24), assuming only that the meshes are locally T-conform.

Numerical experiments

We are now in position to present some numerical illustrations. Consider the case of a squared cavity with a triangular inclusion. Define the square Ω := ( -4; 4) × ( -4; 4): Ω 2 is an equilateral triangle of height 4 (the center of gravity G of Ω 2 is located at (x 1 , x 2 ) =(0, -1/3)) and Ω 1 = Ω \ Ω 2 . We want to approximate problem (1) set in Ω with a constant right-hand side f :

Find u ∈ H 1 0 (Ω) such that: -div(σ∇u) = 1 in Ω .
Denoting (c n ) n=1,3 the corners of Σ, then Ω 1 locally coincides near c n , n = 1, 3 with the cone of aperture α := 5π/3. For this configuration one finds I α := 5, and with theorem 1 one can prove that if κ σ ∈ [-5;-1/5] then problem (1) (with the data f = 1) is well-posed in the Fredholm sense. Then in order to define locally T-conforming meshes, proceeding as in sections 3 and 5, one defines first a neighborhood of the interface Σ made of three polygonal neighborhoods (one for each corner) where one applies the ad hoc symmetry-and rotation-based operator [START_REF] Schatz | An observation concerning Ritz-Galerkin methods with indefinite bilinear forms[END_REF], and three trapezoid-based neighborhoods (one for each part edge) where one applies the symmetry-based operator [START_REF] Smith | Metamaterials and negative refractive index[END_REF]. The associated locally T-conforming mesh is deduced by meshing one pattern of each neighborhood, and duplicating it by symmetry (see figure 8 (right)): in this configuration we choose p 1 = p 2 = p 3 = 10 and q 1 = q 2 = q 3 = 2.

- In figure 9, we plot the relative errors in L 2 -norm for several standard meshes (figure 8 (left)) and several T-conforming meshes (figure 8 (right)), for a contrast κ σ = -5.2, and for several orders of approximation (d = 1, 2, 3). Note that we do not compare the discrete solution with the exact solution. As the optimality of the approximation (see theorem 2) with T-conforming meshes is satisfied, one can compare with u hmin the discrete solution obtained on the finest mesh (with the same order of approximation) and one can check in figure 9 that convergence is ensured with a monotonic behavior of the error. With standard meshes however, the simulation gives unsatisfactory results with a slightly erratic behavior of the error. Numerical results illustrate the importance of using T-conforming meshes to guarantee convergence of the numerical method. Due to the symmetry of the geometry (and the right-hand side), one is expecting a symmetric solution with respect to Ox 2 . One can check that, even for a refined, but standard mesh, the discrete solution does not satisfy this symmetry principle (see figure 10). This also explains why the approximation considering standard meshing does not converge so well. On the other hand, according to [START_REF] Chesnel | T-coercivity and continuous Galerkin methods: application to transmission problems with sign changing coefficients[END_REF]Proposition 2] the regularity of the solution u of problem ( 1) is such that, away from the corners, u i , i = 1, 2 is piecewise-H 2 whereas near the corners there exists a singularity exponent s ∈ (0, 1] such that u i is only guaranteed to be piecewise-H 1+s due to the presence of a singular part here (cf. section 3). Then one has the estimates

u -u h V ≤ C h s , and u -u h Ω ≤ C h 2s .
The last estimate is obtained with the so-called Aubin-Nitsche lemma [START_REF] Ern | Theory and practice of finite elements[END_REF]. Note that in absence of the singular part, one expects that s = 1. For κ σ = -5.2 computations yield a singularity exponent s = 0.13. The results of figure 9 show an average order of convergence equal to 0.9 (that is s = 0.45) while using T-conforming meshes: this super-convergence phenomenon is classically observed when the reference solution is also a computed one, and both solutions are very close one to the other. On the other hand, using standard meshes, the average order of convergence for the last two meshes is s = 0.11, which is almost equal to the theoretical value.

Concluding remarks

In this paper we extended the approximation theory for transmission problems with sign-changing coefficients with the T-coercivity approach. The novelty lies in the treatment of polygonal interfaces. This construction can be directly applied to the discrete problem by designing T-conforming meshes, which allows one to guarantee convergence of conforming finite element methods as soon as the continuous problem is well-posed. Below, we list some possible applications of our work.

• One can study the transmission problem in the time-harmonic regime:

Find u ∈ H 1 0 (Ω) such that: div(σ∇u) + ω 2 ςu = f in Ω , (31)
at the frequency ω = 0, with ς ∈ L ∞ (Ω). Indeed, the added term ω 2 ςu is a compact perturbation, so one concludes that when κ σ ∈ [-I α ; -1/I α ], problem (31) is well-posed in the Fredholm sense and, when it is well-posed, one obtains convergence of the cFE method using locally T-conforming meshes.

• One can extend the results on problems ( 1) and (31) to the transmission problem with sign-changing coefficients and boundary conditions other than Dirichlet boundary condition (see for instance [10, Chapter 1, §1.7]).

• The previous proofs can be easily adapted to 2D geometries with a curved boundary or a curved interface. In particular, the case of the curved interface can be covered with the help of Theorem 3.1-1 of [START_REF] Ciarlet | Mathematical Elasticity[END_REF]. For an interface without corners, well-posedness in the Fredholm sense is then established as soon as κ σ = -1 (see [START_REF] Carvalho | Étude mathématique et numérique de structures plasmoniques avec des coins[END_REF]). We refer to [START_REF] Costabel | A direct boundary integral method for transmission problems[END_REF] for the first relation -and proof with the help of integral equations -of this result. Results for the optimality of the approximation hold using for instance isoparametric quadrilateral FE (see [START_REF] Carvalho | Étude mathématique et numérique de structures plasmoniques avec des coins[END_REF]).

• One may apply similar techniques to a transmission problem with sign-changing coefficients in 3D geometries with straight or curved boundary and interface. However the ability of the T-coercivity approach to deliver the optimal condition on the contrast has yet to be established in the general case [START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF].

• Once the well-posedness of problem ( 31) is established, one can solve the eigenvalue problem

Find (u, λ) ∈ H 1 0 (Ω) \ {0} × C such that: -div(σ∇u) = λu in Ω , (32)
and derive classical error estimates for the eigenvalues' approximation (see [START_REF] Carvalho | Étude mathématique et numérique de structures plasmoniques avec des coins[END_REF][START_REF] Carvalho | Eigenvalue problems with sign-changing coefficients[END_REF]).

• Let us mention that a posteriori error estimates for diffusion problems with sign-changing coefficients have been carried out in [START_REF] Ciarlet | Robust a posteriori error control for transmission problems with sign-changing coefficients using localization of dual norms[END_REF] and lead to similar meshing requirements near the interfaces.

• On the other hand, the study of problem (1) when κ σ belongs to the critical interval cannot be handled as previously, due to the appearance of singularities [START_REF] Ramdani | Lignes supraconductrices : analyse mathématique et numérique[END_REF][START_REF] Bonnet-Ben Dhia | Analyse spectrale et singularités d'un problème de transmission non coercif[END_REF] that do not belong to H 1 (Ω). This problem has been investigated in [START_REF] Bonnet-Ben Dhia | Radiation condition for a non-smooth interface between a dielectric and a metamaterial[END_REF] for particular geometries: it requires a new functional framework explicitly taking into account the singularities. For a general polygonal interface, a new numerical approximation method is proposed in [START_REF] Bonnet-Ben Dhia | On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients[END_REF], which relies on the use of Perfectly Matched Layers at the corners.

1) in Ω q+1-k 2 , k = 1, i: apply S 1 to v k 1 ; 2) in Ω p+1-k 2 , k = i, p: apply S 2 to v k 1 ; 3) in Ω l 2 , l ∈ I := p + 2 -i, q -i : from l = p + 2 -i to l = q -i, apply R i-(p+l) to v i 1 , then S 2 • R p+1-i-(l+1) to v i
1 , update l → 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

R 2i-(p+q) in Ω q+1-i 2 , which ensures continuity of R adm i v 1 on ∂Ω q+1-i 2 ∩ ∂Ω q-i 2 . Ω i 1 Ω 1 1 Ω 2 1 Ω q 2 v 1 1 v 2 1 e 1 e 2 S 1 v 1 1 v i 1 v p 1 S 1 v 2 1 S 1 v i 1 S 2 v p 1 S 2 v p-1 1 S 2 v i 1
Step 2

Step 1

Step 3 One constructs p admissible operators (R adm i ) i=1,p from V 2 to V 1 similarly, the i-th operator being obtained by:

R 2(i-(p+1)) v i 1 S 2 • R -2 v i 1 R 2i-(p+q) v i 1 Ω p+1-i 2 Ω q+1-i 2 Ω 2 2 Ω 1 2
4) in Ω k 1 , k = 1, min(i, p -1): apply S 1 to v q+1-k 2 ; 5) in Ω k 1 , k = min(i + 1, p), p: apply S 2 to v p+1-k 2 ; 6) in Ω i 1 , if i < p, add up the remaining contributions (v l 2 ) l∈I , with I := p + 1 -i, q -i : for l = p + 1 -i to l = q -i, start by S 2 • R l-(p+1-i) then apply -R (l+1)+p-i , update l → l + 2 and so on. If i = p one adds up (v l
2 ) l∈I , where I = 2, q -p + 1 : for l = 2 to l = q -p + 1 start with -R l then apply the rotation+symmetry-based operator S 1 • R (q-p+1)-(l+1) , update l → 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 Ω 1 , already considered at step 4) or 5), and remark that when the index is zero, R 0 = 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 α = 4π/5).

Then one finds four admissible operators from V 1 to V 2 (each one corresponds to a column):

R adm 1 v 1 R adm 2 v 1 R adm 3 v 1 R adm 4 v 1 S 2 v 4 1 , S 2 v 4 1 , S 2 v 4 1 , S 2 v 4 1 , in Ω 1 2 S 2 v 3 1 , S 2 v 3 1 , S 2 v 3 1 , R -2 v 4 1 , in Ω 2 2 S 2 v 2 1 , S 2 v 2 1 , R -4 v 3 1 , S 1 v 4 1 , in Ω 3 2 S 2 v 1 1 , R -6 v 2 1 , S 1 v 3 1 , S 1 v 3 1 , in Ω 4 2 R -8 v 1 1 , S 1 v 2 1 , S 1 v 2 1 , S 1 v 2 1 , in Ω 5 2 S 1 v 1 1 , S 1 v 1 1 , S 1 v 1 1 , S 1 v 1 1 , in Ω 6 2
In this example, |I| = 1 so one applies only one rotation-based operator (following step 3)).

Conversely, one finds four admissible operators from

V 2 to V 1 : R adm 1 v 2 R adm 2 v 2 R adm 3 v 2 R adm 4 v 2 S 1 v 6 2 -R 8 v 5 2 + S 2 v 4 2 , S 1 v 6 2 , S 1 v 6 2 , S 1 v 6 2 , in Ω 1 1 S 2 v 3 2 , S 1 v 5 2 -R 6 v 4 2 + S 2 v 3 2 , S 1 v 5 2 , S 1 v 5 2 , in Ω 2 1 S 2 v 2 2 , S 2 v 2 2 , S 1 v 4 2 -R 4 v 3 2 + S 2 v 2 2 , S 1 v 4 2 , in Ω 3 1 S 2 v 1 2 , S 2 v 1 2 , S 2 v 1 2 , S 2 v 1 2 -R 2 v 2 2 + S 1 v 3 2 , in Ω 4 1
Here |I | = 2 so one applies two additional operators to v l 2 , l ∈ I (following step 6)). With these guidelines one can write all operators (R adm i ) i=1,p , (R 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 Ω 1 with Ω 2 ), one addresses similarly the case q < p. In the following, we set p < q. As mentioned in section 3, the problem is that, taken individually, no admissible operator

R adm i (resp. R adm i ), i = 1, p, satisfies R adm i 2 = I α (resp. R adm i 2 = I α ), with I α defined in (9). Indeed, for all v 1 ∈ V 1 , v 2 ∈ V 2 , for i = 1, p, R adm i built from 1)-2)-3) and R adm i built from 4)-5)-6), one gets the bounds ∇(R adm i v 1 ) 2 Ω2 ≤ k=1,p ∇v k 1 2 Ω k 1 + (|I| + 1) ∇v i 1 2 Ω i 1 ≤ (q -p + 1) ∇v 1 2 Ω1 , ∇(R adm i v 2 ) 2 Ω1 ≤ l∈ 1,q \I ∇v l 2 2 Ω l 2 + (|I | + 1) l∈I ∇v l 2 2 Ω l 2 ≤ (q -p + 1) ∇v 2 2 Ω2 .
The bounds are sharp since:

• in the first case, one may choose

v 1 = 0 such that v k 1 = 0 for k = i: then ∇(R adm i v 1 ) 2 Ω2 = (q -p + 1) ∇v 1 2 Ω1 ; • in the second case, given l ∈ I ( = ∅), one may choose v 2 = 0 such that v k 2 = 0 for k = l: then ∇(R adm i v 2 ) 2 Ω1 = (q -p + 1) ∇v 2 2 Ω2 .
One can check that I α < q -p + 1 for all 1 < p < q: I α -(q -p + 1) = (p -q)(p -1)/p < 0( 9 ). To get optimal operators (that is of norm equal to I α ), one defines R, R as the average of the admissible operators:

R = 1 p i=1,p R adm i , and R = 1 p i=1,p R adm i . ( 33
)
9 Equality is obtained for p = q (corresponding to a plane interface) for which R adm Let us give the general expression of the matrix M for any p, q, and evaluate M M 2 to conclude. For Remark 5. Following remark 4, one can find w 0 1 ∈ V 1 such that ξ 1/2 ∇(Rw 0 1 ) 2 Ω2∩S ≥ ξ 1/2 ∇w 0 1 2 Ω1∩S .

n ∈ 1, p , define M p-n+1 , M p-n+1 ∈ M p-n+1,p (R) such that M p-n+1 M p-n+1              0
Proposition 5. Let ξ be a smooth positive function with support S that is symmetric with respect to the interface. Then for all

w 1 ∈ V 1 ξ 1/2 ∇(Rw 1 ) 2 Ω2∩S = ξ 1/2 ∇w 1 2 Ω1∩S ,
where R is the symmetry-based operator [START_REF] Smith | Metamaterials and negative refractive index[END_REF].

Proof. This inequality is obtained using the change of variables (x Σ , y Σ ) → (x Σ , -y Σ ) in Ω 2 ∩ S. For all

w 1 ∈ V 1

A.3 Local and global interpolation estimates

Let τ be the reference triangle, with corners (1, 0), (0, 1) and (0, 0), and let Îd be the Lagrange interpolation operator over P d (τ ). Given (T h ) h a regular family of triangulations of a domain Ω we call I d τ the Lagrange interpolation operator over P d (τ ), for all h and all τ ∈ T h . )] 1/2 . Then, we remark that the 2 -norm of the coefficients is a norm over P d (τ ), hence it is also a norm over its vector subspace P zmv d (τ ) made of zero-mean value polynomials on τ . Now, over P zmv d (τ ), the semi-norm ∇ • τ is also a norm and because P zmv d (τ ) is a finite dimensional vector space, both norms are equivalent. Noting finally that, starting from v, one has only to modify the degree-0 coefficient to obtain a zero-mean value polynomial, we finally get that there exists Ĉ independent of v such that ( i=0,d |a i | 2 ) 1/2 ≤ Ĉ ∇v τ . Taking Ĉ = Ĉd Ĉ leads to (35). We recall that provided the family of triangulations (T h ) h is regular, there exists s > 0 such that, for all h and for all τ ∈ T h , there holds h τ ≤ s ρ τ , where ρ τ is the diameter of the largest ball that can be inscribed in τ . One can then derive (36) from (35) by using the affine mapping from the reference triangle τ to the triangle τ ∈ T h . We refer for instance to [START_REF] Ern | Theory and practice of finite elements[END_REF]. We report here the computations for the sake of completeness (C 0 , C 1 , • • • are constants that are independent of h and τ ): Let us prove the estimate over Ω h 2 = int( p=1,P ∪ τ ∈T p h,2 τ ). Recall that V := H 1 0 (Ω), V h (d) := {v ∈ V : v| τ ∈ P d (τ ), ∀τ ∈ T h }, and I h is the interpolation operator on V h (d) .

Lemma 5. Consider a cut-off function χ, and denote by χ h its interpolation of degree 1. For all v h ∈ V h (d) ,

∃C > 0, ∇(χ h v h -I h (χ h v h )) Ω h 2 ≤ C h χ W 1,∞ (Ω2) ∇v h Ω2 . (37)
Proof. To obtain (37), we evaluate the L 2 -norm of ∇(χ h v h -I h (χ h v h )) on Ω h 2 by splitting Ω h 2 into triangles, and then going back to the reference triangle to use the uniform estimate (36):

∇(χ h v h -I h (χ h v h )) 2 Ω h 2 = τ ⊂Ω h 2 ˆτ |∇(χ h|τ v h |τ -I d τ (χ h|τ v h |τ ))| 2 ≤ C τ ⊂Ω h 2 h 2 τ |χ h|τ | 2 W 1,∞ (τ ) ∇v h |τ 2 τ .
Using the definition of the meshsize h yields

∇(χ h v h -I h (χ h v h )) Ω h 2 ≤ C h χ h W 1,∞ (Ω2) ∇v h Ω2 .
One concludes using the stability estimate (25).

Figure 1 :

 1 Figure 1: Examples of geometries that have been handled with the T-coercivity: (left) a square where Ω 1 is one quadrant ; (right) a disk where Ω 1 is an angular sector of angle α.

Figure 2 :

 2 Figure 2: Examples of pattern-based geometries with a corner c of angle α = 2π/3 measured in Ω 1 : (left) sector-based ; (middle) triangle-based ; (right) leaf-based geometry.

Figure 3 :

 3 Figure 3: Representation of the two admissible geometry-based operators R.

Figure 4 :

 4 Figure 4: Representation of thetwo admissible geometry-based operators R adm : (left) one applies an axial symmetry in Ω 1 2 to reach Ω 2 1 and folds the rest of v 2 into Ω 1 1 to ensure continuity ; (right) one applies an axial symmetry in Ω 4 2 to reach Ω 1 1 and folds the rest of v 2 into Ω 2 1 to ensure continuity.

Figure 5 :

 5 Figure 5: (Left) standard mesh (h = 0.2, 385 nodes), Ω hex 1 corresponds to blue region while Ω hex 2 corresponds to the green one ; (right) T-conforming mesh (h = 0.2, 379 nodes).

Figure 6 :

 6 Figure 6: (Left) relative error in H 1 -norm for different mesh sizes h (log-log scale) for κ σ = -3 ; (right) relative error in H 1 -norm for different mesh sizes h (log-log scale) for κ σ = -2.1.

Definition 1 .

 1 Let b(•, •) be a continuous sesquilinear form on E × E. It verifies a weak stability condition if

Remark 1 .Lemma 1 .

 11 When β ≤ 0 in (13), one recovers the classical stability condition. Respectively, when β ≤ 0 in (14), one obtains T-coercivity for b(•, •). The operator T introduced in (14) realizes the inf-sup condition (13): it is sometimes called an inf-sup operator. One can easily prove the following result, see e.g. [8, Chapter 2]. Let b(•, •) be a continuous, sesquilinear hermitian form on E × E. Then the three assertions below are equivalent: (i) (11)-(12) is well-posed in the Fredholm sense ; (ii) the form b satisfies a weak stability condition ; (iii) the form b is weakly T-coercive.

Figure 7 :

 7 Figure 7: Notations around two corners c 1 and c 2 : here p 1 = 2, q 1 = 6, p 2 = 11 and q 2 = 3. The gray zones represent the support of χ p while the empty-dashed domains represent the support of χ N +p , p = 1, N .

Figure 8 :

 8 Figure 8: (Left) example a standard triangulation of Ω h (1430 nodes) ; (right) example of locally T-conforming mesh (1424 nodes). The dashed lines enclose regions where we built T-conforming meshes near a corner, respectively a straight line.

Figure 9 :

 9 Figure 9: Relative error in L 2 -norm of u h -u hmin for different mesh sizes h (log-log scale) for κ σ = -5.2.

Figure 10 :

 10 Figure 10: Computed solution, same color scale: (left) solution u h obtained with P3 finite elements on the finer standard mesh (101576 nodes) ; (right) solution u h obtained with P3 finite elements on the finer T-conforming mesh (101716 nodes).

Figure 11 :

 11 Figure 11: Scheme representing steps 1)-2)-3) for building R adm i .

i 2 =

 2 R adm i

k 1 (r k , ϕ k ) ∂ϕ k 2 r k dr k dϕ k 1

 121 . . . . . . . . . . . . . . . . . .

ξ 1 / 2 ∇

 12

Lemma 4 .| 2  1 / 2 . 2 τ + ∇ fi 2 τ

 421222 On the reference triangle τ , one has the estimate∃ Ĉ > 0, ∀ χ ∈ P 1 (τ ), ∀v ∈ P d (τ ), ∇ χv -Îd ( χv) τ ≤ Ĉ | χ| W 1,∞ (τ ) ∇v τ . (35)As a consequence, one has the uniform local estimate∃C > 0, ∀h, ∀τ ∈ T h , ∀χ ∈ P 1 (τ ), ∀v ∈ P d (τ ), ∇ χv -I d τ (χv) τ ≤ C h τ |χ| W 1,∞ (τ ) ∇v τ . (36) Proof. Write χ(x, ŷ) = α + χ1 (x, ŷ), resp. v(x, ŷ) = v-(x, ŷ) + vd (x, ŷ) where χ1 (x, ŷ) = β x + γ ŷ, resp. vd (x, ŷ) = i=0,d a i xi ŷd-i and deg(v -) < d. For i = 0, d, define êi (x, ŷ) = xi+1 ŷd-i -Îd (x i+1 ŷd-i ), resp. fi (x, ŷ) = xi ŷd+1-i -Îd (x i ŷd+1-i ). Note that | χ| W 1,∞ (τ ) = max(|β|, |γ|).Since there holds χv -Îd ( χv) = χ1 vd -Îd ( χ1 vd ) = β i=0,d a i êi (x, ŷ) + γ i=0,d a i fi (x, ŷ), one finds∇ χv -Îd ( χv) (x, ŷ) ≤ |β| i=0,d |a i | | ∇ê i (x, ŷ)| + |γ| i=0,d |a i | | ∇ fi (x, ŷ)| ≤ | χ| W 1,∞ (τ ) i=0,d |a i | | ∇ê i (x, ŷ)| + | ∇ fi (x, ŷ)| ≤ | χ| W 1,∞ (τ ) ∇ê i (x, ŷ)| + | ∇ fi (x, ŷ)| It follows that ∇( χv-Îd ( χv)) τ ≤ Ĉd | χ| W 1,∞ (τ ) ( i=0,d |a i | 2 ) 1/2 , with Ĉd = √ 2[ i=0,d ( ∇ê i

∇

  χv -I d τ (χv) τ ≤ C 0 s ∇ χv -Îd ( χv) τ cf. Lemmas 1.100-1.101 of [16] ≤ C 1 | χ| W 1,∞ (τ ) ∇v τ cf. (35) ≤ C 2 h τ |χ| W 1,∞ (τ ) × C 3 ∇v τ cf. Lemmas 1.100-1.101 of[START_REF] Ern | Theory and practice of finite elements[END_REF], that is (36) with C = C 2 C 3 .

From now on, we denote by • O the L

-norm over the open set O.

It is an attempt only, because there is a mistake in the definition of the discrete operator T h that will be clarified in section 5.

With this choice ur(•, θ) = 0 for θ ∈ {nβ, n ∈ Z}, then continuity is easily ensured at the crossing of the interfaces.

B 2N is composed of two trapezoids of side lengths s 2N and s 1 .

The strictly positive, upper limit distance up to which this property applies is a function of minp ρ min,p and of (αp) p=1,N .

In accordance with the previous notations, one chooses ρ min,p such that ρ min,p < ρc p .

A Discontinuous Galerkin approach has been studied in[START_REF] Chung | A staggered discontinuous Galerkin method for wave propagation in media with dielectrics and meta-materials[END_REF].

= 1, or p = 1: in that case there is only one admissible operator.

A Appendix

A.1 General construction of R around corners

In this section we generalize the tilings method presented in section 3 to any corner of angle α ∈ 2πQ. Recall that we define α = 2π p p+q in Ω 1 , with p, q > 0, p = q and p + q even. Proceeding as in section 3, one builds admissible rotation-and symmetry-based operators R adm , and then take the average of these admissible operators to obtain the desired result (that is operators R and R with the same minimal norm as in [START_REF] Bonnet-Ben Dhia | T-coercivity for scalar interface problems between dielectrics and metamaterials[END_REF]). We propose min(p, q) admissible operators below based on a simple algorithm. Consider for instance that p < q. One constructs p admissible operators (R adm i ) i=1,p from V 1 to V 2 , the i-th operator being obtained by (see figure 11 for an illustration): Going back to our example with p = 4 and q = 6, one defines R and R such that for all

These operators are optimal in the sense that R 2 = R 2 = I α . Proceeding as in section 3, using the triangle inequality one finds that, for all

.

By defining the matrix

, and

Let us remark that the entries M lk , l = 1, 6, k = 1, 4, are such that

One obtains equality by taking, for every k = 1, 4, v 1 such that v i 1 = 0 for i = k, which yields

Above, a sum over k appears due to the fact that for l = 1, q, the (R

, where j denotes the column index. In the specific case where n = 1, the matrix M p (resp. 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 1 p , except on one diagonal where they range from n p to p p = 1. Then the shape of M depends on whether p ≤ q

2

or not:

) and respectively M m ∈ M m,p (R) whose entries are all equal to 1/p, with the convention that if m = 0, M m is empty. The example treated in section 3 corresponds to that case with m = 0.

with M p-m+1 , M p-m+1 ∈ M p-m+1,p (R) (defined above for n = m) and resp. M m-2 ∈ M m-2,p (R) such that (below i denotes the row index)

one can obtain some "central" columns with only 1 p entries, whereas for d ≤ 1, one gets central columns of the form (0, . . . , 0, m-j p , 1 p , . . . , 1 p , 1+i p , 0, . . . , 0) and (0, . . . , 0, 1+j p , 1 p , . . . , 1 p , m-i p , 0, . . . , 0) 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 ( m-1 p , 1 p , . . . , 1 p , m-2 p , 0) , (0, m-2 p , 1 p , . . . , 1 p , m-1 p ) and for d = 1 the central column is

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 = 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 ≤ 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

, and

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

, and

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

R) satisfy the following properties:

Proposition 2. The matrix M ∈ M q,p (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 [START_REF] Bonnet-Ben Dhia | On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients[END_REF], i.e. when q ≥ p. First consider the case where q = 2p + m, m ∈ 1, p . Then for all k = 1, p one has

Now let us consider the case q = 2p -m, m ∈ 3, p -1 and suppose that d > 1:

.

One reaches the same result in the specific cases d ≤ 1 by noticing that for l = 1, p -m + 1, (M p-m+1 ) lk = ( M p-m+1 ) lk = 0 for k = m -2 + d, m -1.

It follows that

Proposition 3. For all p, q > 0 let M ∈ M q,p (R) be a matrix which satisfies the properties of proposition 2. Then M M 2 = max q p , p q .

Proof. Define A := M M , it holds A 2 = λ max = max Due to the second and the third properties in proposition 2 satisfied by M , one finds

Let us prove that λ max = max p q , q p . Consider the vector W ∈ R p such that W = (1, . . . , 1) . Suppose for instance that p ≤ q, then using again the second and the third properties in proposition 2, we get for all j = 1, p

Consequently, the proposed operator R in (33) is of optimal norm. One proceeds similarly for R by considering M 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 k0l0 ), one readily checks that if w 0

A.2 Weighted estimates for operators R

In §A.1 we provided bounds for the norms of the geometry-based operators R. Here we provide bounds for the operator norm when we use a localization process (see section 4), that is when the operator R is locally applied in the neighborhood of the interface thanks to a cut-off function ξ (defined as in section 4) whose support is localized either near a corner (proposition 4) or a straight line (proposition 5) of the interface Σ. We use the same notations as in §A.1. Proposition 4. Let ξ be a smooth positive function with support S, that depends only on the distance to the corner of angle α = 2π p p+q . Then

,

where I α = max p q , q p , and R is a rotation-and symmetry-based operator from V 1 to V 2 defined as in [START_REF] Schatz | An observation concerning Ritz-Galerkin methods with indefinite bilinear forms[END_REF]. Then similar to (34) with the change of variables (ρ, θ) → (r k , ϕ k ) for k = 1, p that maps Ω l 2 to Ω k 1 (note that