Gaelle Petit

Gwenola Yannou-Le Bris

Gilles Trystram

CODESIGN OF SUSTAINABLE PERFORMANCE OBJECTIVES IN FOOD A VALUE CHAIN

INTRODUCTION

Integrating sustainable perspective in the design process of products/services derived from living systems is a more and more important matter. (Melanie Fritz and Gerhard Schiefer, 2009). Food sector is at the center of stakes easily perceived: availability of food resources for all populations and communities (Techane Bosona and Girma Gebresenbet, 2013), respect for animal welfare (Beth A. [START_REF] Ventura | Animal Welfare Concerns and Values of Stakeholders Within the Dairy Industry[END_REF] but also respect for eating habits (Xavier Irz et al., 2013). The complexity of the situations under consideration is intensified by the global trade internationalization and the prices volatility which reinforce the instability for organizations of producers and agricultural resources buyers (Jacques H. [START_REF] Trienekens | Transparency in complex dynamic food supply chains[END_REF]. In this context for several years the literature review shows an increasing number of publications on the emergence of food value chains (FAO and David Neven, 2014). Value chains are to be differentiated from the concepts of supply chain (Laurent Trognon, 2009). The term "supply chain" is commonly used to designate relations of products and services flows exchanges between companies and commercial actors. The value chain concept implies the willingness of the supply chain actors to set up durable relationships and to develop various forms and intensities of collaboration. This paradigm of value chains makes sense when it comes to increasing the sustainability of end products to be offered to the market (Peter Gibbon and Stefano Ponte, 2008). Indeed, many studies have shown that the final sustainability of a food product is largely defined by the upstream production stages (J. [START_REF] Humphrey | Global value chains in agrifood sector[END_REF][START_REF] Humphrey | Global value chains in agrifood sector[END_REF]. However, the economic pressures exerted by international competition have reduced the economic capacity of upstream food producers to build and improve the sustainability of food production. In a context of intense competition inherent to food markets in developed countries, the idea of increasing collaboration between actors from upstream and downstream in order to offer a more sustainable food offer is highly praised (Tzu-An Chiang and Amy J. C. [START_REF] Chiang | Development of value chain collaborative model for product lifecycle management and its LCD industry adoption[END_REF]. The question arises of establishing the nature and intensity of sustainability performance that the actors will seek to achieve together. If the assessment of the environmental performance of a product is already by itself multi-criteria (Damien Craheix et al., 2012;Caroline Sablayrolles et al., 2014) assessing its sustainable performance increases operation complexity. Indeed the number of criteria that have be taken into account to establish the sustainable performance (including environmental, economic and social criteria) of a food offer (since to environmental indicators are added economic and social indicators) (Claudia R. [START_REF] Binder | Considering the normative, systemic and procedural dimensions in indicator-based sustainability assessments in agriculture[END_REF] is not only augmented. It induces different perimeters and temporalities considered for the systems analyzed. Yet if the actors of a value chain want to build together a more sustainable value chain, they must also evaluate their level of performance and define together how they will improve it. In the research works presented here the question of actors' points of view convergence for the assessment and design of this sustainable value chain is raised. This work refers to a class of issues that constitute collective action: decision, rationalization, representation, legitimacy, cooperation, prescription (Albert David et al., 2012). Given each actors expectations and constraints, and thus the variability in the emphasis placed on the various sustainability criteria, is it possible to create solutions from designing the value chain that satisfy all the actors?

METHOD AND PRESENTATION OF MODELS USED

The selected value chain for this experimentation is a chain of pork production. The animals are bred, slaughtered and the meat is distributed in France. The perimeter considered here extends from breeding to consumers sales.

Assessed models

Through an analysis of the scientific literature and expert interviews, the major sustainability issues associated to the value chain are identified: the sustainability hotpoints. These hotpoints are potential impacts (environmental, social and economic) for the stakeholders (representatives of the environment defense, experts, society). The analysis of the activities carried out during the product life cycle leads to assume strong causal links between the sustainability performances of the sector and:



The composition of the animal feed ration; 

The preparation method of the animal feed ration; 

The managing method of material losses; 

The place of meat cutting.

In the case of feed preparation, differentiating factors considered are the type of preparation from the feed and the dominant raw material. Indeed, there is a great variability in the way the food is prepared, from the farmer who produces all the raw material on the spot to the farmer who buys the whole of his food after an Animal nutrition cooperative. All intermediary cases are possible. Also extreme situations are represented in this work: a breeder who produces all the feed (except the mineral supplement), a breeder who buys all the feed and a breeder which represents a mix between the two, that is, it buys part of the food and produces another part of it. On the other hand, these breeders and cooperatives do not all use the same cereal or the same dominant protein. The choice depends on the nutritional balance of the ration, the cost of the raw materials and their availabilities. The major trends are represented here: maize, soybean or rapeseed dominance according to the methods of preparation.

On the other hand, the shared use of a methaniser between the actors of the value chain induces reuse of wastewater from livestock production, slaughter waste at the slaughterhouse and at the level of the manufacturing plant and waste of various kinds at the store level (food oils, bread and biscuits, meat and meat products, dairy products). This implies a regional implementation of the value chain that is being studied here. Indeed as transport distances are proportional to environmental impacts and costs these transports must be minimized. The electricity produced by this methanizer is re-injected into the livestock farm as well as into the slaughterhouse, enabling the model to operate in a closed loop for part of the electricity. Finally, the place of second cutting of the meat refers to two alternatives: one, existing, takes place in the factory of cutting after slaughter and transport of the lanyards of pig. The punched pieces (roast, sausages, ribs, etc.) are then transported packed to the distribution shop. The second, theoretical, implies that the pork loins are directly transported to the store and cut on the spot in a traditional butcher shop. The actors of the value chain (producer, distributor) validated the interest to assess impacts on sustainability of these alternative solutions. To conduct these assessments several combinations of scenarios have been tested. The reference model describes a scenario "Buying a complete feed from an animal nutrition cooperative, soy predominant" (scenario D). Ten other scenarios were constructed (outlined below) and evaluated (see next section).

Assessment of sustainable performance for each model

In order to evaluate the sustainable performance of each scenario, three types of indicators are used: environmental, social and economic. Environmental indicators are calculated using LCA software. The values of the other indicators are derived from the bibliography (especially for raw material costs, transaction between actors or feed preparation facilities operation). The values of the social criteria are also derived from the professional publications or social studies identified in the sector. As a result, the selected indicators do not cover all the social and economic hotpoints recommended to be handled, but those for which data exist, are accurate and traceable. The sensitivity to the scenarios tested was the last criteria retained for selecting the socio-economic data used for the scenario evaluation. As a very large majority of the indicators are impacts, the objective for the actors in a value chain is to minimize their numerical value. Consequently each impact value must therefore be analyzed with "the lower the better" point of view. The set of selected indicators for the experiment is summarized in Table 2.

Indicators weighting by the actors of the value chain

The indicator grid was submitted to some actors in the value chain to define the relative importance they give to each of these indicators. These actors are here considered as "super" experts. Two actors were contributing but their good knowledge of the value chain made it possible to ask them to take the place of more specific actors. On one hand an agricultural cooperative expert answered for farming phase (actor 1) (including breeding, slaughterhouse and transformation). On the other hand the responsible Sustainable Development of the distribution cooperative reported on the opinions of supply (actor 2) and sales activities (actor 3a and 3b). The 3a and 3b distinction comes from the fact that the interlocutor saw two different visions representative of store managers, he then detailed them two. The manager 3a represents the average manager, the most widespread in this type of organization. The manager 3b is more sensitive to the overall sustainability indicators and not only to his economic performance. Initially, each interlocutor was asked to assess the relative importance of the three pillars of sustainable development among themselves. Five scale levels are analyzed: from very high importance (weighting value 1) to very small importance (weighting value 5). Subsequently, the same interlocutors were invited to express their views on the relative importance of the indicators within each pillar. The same scaling factors were applied. These two successive weights were applied to the matrix of the sustainable performance values obtained for the 11 scenarios described in section 2.1. However, to allow the application of these weights the raw performance values of the scenarios have been standardized beforehand. The standardization step has previously asked to carry out a ranking of the scenarios performances between them by indicator (while maintaining the hierarchical order which gives the best solution the smallest numerical value). A synthetic representation of the data processing is given in Figure 1.

RESULTS

Table 3 shows the results by indicator for each scenario. The scenarios are classified as shown in Section 2 from A to K and the indicators are ranked from 1 to 10 per pillar: social, economic, environmental. Colours have been added to the table for greater readability. Green tones correspond to the best-performing scenarios by indicators, so these are the lowest values. Red, on the other hand, corresponds to the worst-performing scenarios and the highest values. There are 11 maximum positions for each scenario; some colours (the darkest green or the darkest red) correspond to two positions (respectively the lowest score and the strongest score). Table 4 shows the weights given by the actors in the value chain in response. They were initially asked for their opinion on the relative importance of the three pillars between them: environmental, economic and social. In the second stage, the actors spoke about the relative importance of indicators within each of the three pillars. The importance of the pillars is given in the vertical boxes for each of the actors: agricultural cooperative, logistician, store 1 and store 2. The weight of the indicators is given in the horizontal boxes for each of these actors. In total, five distinct importance groups appeared and are represented by a colour code from factor 1 to factor 5. Finally, Table 5 gives the final single scores for each of the scenarios studied and for each of the weights proposed respectively by the players in the value chain. Only the three best cases (green, lower values) and the three worst cases (red, higher values) are identified for each of the given weights. In dark green for example, scenario H is the most efficient for actor 1 whereas scenario G is the more efficient for actor 2 and for unweighted calculation, all indicators with equal weight. In dark red, scenario B is the worst-case scenario all actors excepted for actor 1. Scenario F is the second worst-case scenario for all actors excepted for actor 1.

DISCUSSION

The subject of this research is centred on the possibility for the actors of a value chain to define in common criteria to assess the sustainability of their activities. The first result of this experimentation is the difference of weighting given by each actor at the sustainable pillars and criteria (Table 4). For all of them economic criterion is the more important but for agricultural cooperative the difference of weighting between economic criterion and other criteria is stronger. The important economic troubles shared by the agricultural sector explain this particular position.

The results of the table 5 highlight the three best (gradient of green) and the three worst cases (gradient of red) for each actor interviewed. These results show first of all that there is no strong contradiction between the actors. In other words there is no scenario into which the best triplet for one actor is the worst triplet for another actor. Indeed, the scenarios A, B and F (full on-farm production of the feed / with rapeseed or maize dominant and mixed feed with maize dominant) are the worst triplet for all the actors. Scenarios H, J and K (methanizer shared / mixed feed, linseed addition, methanizer shared / mixed feed, linseed addition, methanizer shared, cutting at the distribution shop) are the best triplet with the same ranking for all actors. There is absolutely no contradiction in theoretical improvement solutions, while there are minor contradictions (within the worst triplet) in existing theoretical solutions. It is important to recall here that these rankings were performed without actually presenting the results to the actors. These last provided only the hierarchy of their weighting of indicators and accepted the types of solutions that needed to be assessed.

There actually are cases where some scenarios are unanimous but this apparent consensus conceals a form of complexity that will be detailed in further work. Indeed the three scenario with methanizer shared seem systematically more efficient than all the others but their economic performance remains poor due to the investment required. However given the weighting that was applied according to the actors without having access to the different scenarios performance results, these poor results are not sufficient to disadvantage these scenarios. And yet the individual economic values for initial indicators at the beginning of the data processing are very bad and they could have been expected to severely disadvantage the three considered scenarios. As a result, it is doubtful whether the actors of the value chain would have been interested in these scenarios. In others words, these three scenarios are surely the solutions that best fit the values the actors want to pursue, not necessarily those they can afford. In any case it is clear that this must be verified in additional work. To go further choosing one of these solutions, given its bad economic situation, the results comparison matrix becomes a common reflection frame to question about who "pays" the related investments and truly start to codesign a more sustainable value chain. If the discussed solutions, i.e. sharing a methanizer, clearly improves the global sustainable performance score of the value chain and according to the declared preferences of the actors of this chain, it is a starting point to establish which kind of economic solidarity is to develop between them.

A final remark can be made about grouping solutions by category of actor. Indeed, the results of the second store manager are very close from the unweighted ranking. This is explained by the fact that the choice of this actor leads to the equal consideration of the three pillars (environmental, economic and social) of the selected indicators. These two classifications give an indication of the scenarios chosen when none of the three pillars (and in particular the economic pillar, often widely promoted by the actors) is favored. These solutions, while representing a different equilibrium from that which is considered as the most expected, provide a good basis for reflection on what may be a more sustainable value chain.

The highlighted elements make it possible to answer the question asked. There are, in fact, areas of common satisfaction between the actors and certain scenarios can be selected as performing by several actors or all the actors. On the other hand, to arbitrate more in detail between scenarios that do not constitute extremes or to arbitrate among extremes triplets, discussions and negotiations between actors will be necessary. If the upstream and the downstream of a value chain do not communicate or do not cooperate, the risk that some actors will implement improvement solutions that do not satisfy other players in the value chain is not negligible. This study showed that some scenarios tend to impose themselves by their "spontaneous" performances. It is always interesting to highlight it because these choices can be the subject of a consensus quickly. For other cases, the actor's joint decision seems to be more important. Thus the actors of the same value chain, that is to say actors predisposed to work together on the sustainability of their practices and the products they put on the market and for which they are responsible in the eyes of the consumer, have a real interest in sharing tools, common representations of their global system in order to better understand the arbitrations that can be joined. This is the case for advancement in the increments to be discussed for continuous improvement. These are therefore iterative actions, both from design and management of the sustainability of organizations practices and products. Even if the weights granted by the actors are different, the ranking order of the different scenarios is not so different. Finally, stakeholders could reach agreement and conciliate on the best and worst scenarios to be implemented as part of an enhanced desired sustainability of the value chain. The remarks made above concerning the economic performance of the best solutions (sharing a methanizer) show the importance of the negotiation between actors to allow local arbitrations. The poor results of selected economic indicators penalize certain actors in particular those of the upstream (i.e. breeder). It is possible to propose trade-offs for the four solutions including the sharing of a methanizer. Nevertheless, the field of experimentation chosen here is special since the players in this value chain already know each other and seek to build together more sustainable approaches. We can therefore assume that they already share common values and that this explains the coherent rankings.

It has been shown that a compromise can be constructed, and this result differs from what is usually read in the literature. It is much more frequently a focal actor that gives the lines of conduct to be followed by everyone. Many new questions are coming out from this study. Indeed it would be interesting to measure how and how much results change if treatment stages are modified. Further works on sensitivity study is expected for example looking at the impact of changing some aggregation data elements or discuss whether one of the actors over or under evaluates his weighing.

CONCLUSION

This study has shown that measuring the performance of different solutions, classifying them according to a weighting of pre-established sorting criteria allows decision-makers in a value chain, therefore predisposed to work together, to find a consensus on sustainability alternatives for products and practices. It is important to specify here that the various actors interviewed expressed their respective weighting for the various sorting criteria before the respective performances of the different scenarios were presented to them. According to Simon (James Gardner March and Herbert Alexander Simon, 1965), economic rationality and optimal choice imply that all information (choice possibilities) is available, that the decision-maker can measure the performance linked to each of the possibilities and that he can classify the performance in function of a predetermined order of preference. Within organizations, social actors cannot find an optimal solution because of their cognitive and temporal limitations. The human being can focus his attention on only a limited number of problems and solutions at the same time. To decide an action, he proceeds sequentially and iteratively by comparing the possible solutions to what it considers to be minimum criteria of satisfaction. In general, the decision-makers will stop their choice to the first satisfactory and sufficient solution that emerges. A future study could therefore be the measure of the different choices of the actors when they are aware of the different scenarios performances compared to their blind choice, in other words the sorting criteria weighting before they even know the consequences of their choice.

Table 1 .

 1 The eleven scenarios of animal feeds assessed.

A. Full on-farm processed feed, rapeseed dominant B. Full on-farm processed fee , maize dominant C. Full purchased feed, rapeseed dominant D. Full purchased feed, soy dominant E. Mixed (part on-farm processed and part purchased), soy dominant F. Mixed, maize dominant G. Addition of linseed to the animal feed H. Methanizer shared between actors of the value chain I. Second meat cutting directly at the distribution shop J. Mixed feed, linseed addition, methanizer shared K. Mixed feed, linseed addition, methanizer shared, cutting at the distribution shop

Table 2 .

 2 Indicators and associated references.

	Indicator	Reference	Unity
	Soc1. Carcass pH	Real data from value chain	#
	Soc2. Transport duration	Real data from value chain	h
	Soc3. Foodmiles / localness (local cultures)	Estimation from bilbiography	%
	Soc4. Breeder's welfare (survey)	Estimation from interviews	Score 1 to 5
	Soc5. Employees' welfare (survey)	Estimation from interviews	Score 1 to 5
	Soc6. Biodiversity (number component /formula)	Estimation from bilbiography	#
	Soc7. Sensory evaluation score	Real data from value chain	Score 1 to 10
	Soc8. Omega 6 / Omega 3 ratio	Estimation from bilbiography	%
	Soc. 9. OGM ratio
		Estimation from bibliography Real data from value chain	%
	Eco3. Muscles rate (economical quality)	Real data from value chain	%
	Eco4. Waste (losses)	Estimation from bibliography Real data from value chain	%
	Eco5. Number of hires	Estimation from interviews	#
	Eco6. Additional work hours (-)	Estimation from bilbiography	h
	Eco7. Variation of labor cost	Estimation from bilbiography	€
	Eco8. Short-term investment	Estimation from bilbiography	€/t
	Eco9. Long-term investment	Estimation from bilbiography	€/t
	Eco10. Variation of manufacturing cost per product	Real data from value chain Estimation from bilbiography	€
	Env.1. Climate change		kg CO2 eq
	Env2. Terrestrial acidification		kg SO2 eq
	Env. 3 Freshwater eutrophication		kg P eq
	Env4. Human toxicity		kg 1,4-DB eq
	Env5. Freshwater ecotoxicity Env6. Marine ecotoxicity	Estimation from bilbiography	kg 1,4-DB eq kg 1,4-DB eq
	Env7. Agricultural land occupation		m2a
	Env8. Urban land occupation		m2a
	Env9. Water depletion		m3
	Env10. Fossil depletion		kg oil eq

Real data from value chain % Soc10. Water losses after cooking (Technological quality) Estimation from bilbiography %

Eco1. Additional income paid to breeder

Real data from value chain € Eco2. Production valorization (losses)

Table 3 .

 3 Numerical result of each scenario by indicator.

		A	B	C	D	E	F	G	H	I	J	K
	Soc1	-5,4	-5,4	-5,4	-5,4	-5,4	-5,4	-5,6	-5,4	-5,4	-5,6	-5,6
	Soc2	8	8	8	8	8	8	2,8	8	8	2,8	2,8
	Soc3	15,9	35,0	93,5	91,0	48,5	84,9	91,0	91,0	91,0	15,9	15,9
	Soc4	-2	-2	-2	-2	-2	-2	-5	-2	-2	-5	-5
	Soc5	-2	-2	-2	-2	-2	-2	-5	-2	-2	-5	-5
	Soc6	-9,7	-6,3	-6,0	-5,3	-11,3	-7,0	-6,3	-5,3	-5,3	-9,7	-9,7
	Soc7	-6,02	-6,02	-6,02	-6,02	-6,02	-6,02	-6,34	-6,02	-6,02	-6,34	-6,34
	Soc8	10,7	10,7	10,7	10,7	10,7	10,7	3,8	10,7	10,7	3,8	3,8
	Soc9	7,5	7,5	7,5	7,5	7,5	7,5	0,9	7,5	7,5	0,9	0,9
	Soc10	15,45	15,45	15,45	15,45	15,45	15,45	13,2	15,45	15,45	13,2	13,2
	Eco1	0	0	0	0	0	0	-10	0	0	-10	-10
	Eco2	1	5	5,9	5,9	1	5	4,55	5,9	5,9	1	1
	Eco3	-60,9	-60,9	-60,9	-60,9	-60,9	-60,9	-61,9	-60,9	-60,9	-61,9	-61,9
	Eco4	1	5	5,9	5,9	1	5	4,55	5,9	5,9	1	1
	Eco5	0	0	0	0	0	0	-13	0	0	-13	-13
	Eco6	0,17	0,25	0	0	0,335	0,335	0	0,5	0	0,5	0,5
	Eco7	3	4,5	0	0	6,25	6,25	0	0	0	3	3
	Eco8	24	24	0	0	18	18	100	20	0	144	144
	Eco9	8	8	0	0	6	6	0	0	0	14	14
	Eco10	30	39	0	0	31,3	38,5	0	43,5	0	73,5	73,5
	Env1 1 815 474 1 802 227 1 803 321 1 819 355 1 809 714 1 812 392 1 868 506 128 418 1 856 315 128 763 125 533
	Env2	8 882	8 727	9 047	9 114	9 204	9 432	10 551	5 023	9 626	6 045	6 031
	Env3	81	79	50	59	62	93	118	53	59	114	114
	Env4 2 433 286 2 421 922 2 240 300 2 270 810 2 157 413 2 199 149 2 288 051 220 041 2 267 413 104 266	83 417
	Env5	3 021	3 106	2 592	2 462	2 707	3 157	2 571	762	2 499	1 066	1 044
	Env6	916 151	904 334	776 651	734 365	707 373	712 825	760 476	101 246	744 318	72 952	55 471
	Env7	501 602	415 184	236 829	325 076	318 773	274 965	427 068	325 076	314 857	422 840 422 692
	Env8	614	660	368	192	210	210	216	192	139	203	118
	Env9	29 715	29 002	12 639	9 973	14 345	26 875	9 843	3 170	9 560	7 449	7 432
	Env10 549 687	549 122	550 504	552 019	547 919	552 210	567 459	18 905	565 505	16 379	15 292

Table 4 .

 4 Weighting given by the actors of the value chain

	Actor 1	Actor 2	Actor 3a	Actor 3b

Table 5 .

 5 Aggregated indicators value considering all different data acquisition and processing stages

		A	B	C	D	E	F	G	H	I	J	K
	Actor 1	35,01	34,87	31,68	31,76	28,60	34,47	32,88	23,99	31,76	20,12	16,85
	Actor 2	40,32	39,64	33,08	34,66	32,54	38,07	34,68	27,13	33,90	24,31	20,68
	Actor 3a	30,64	30,26	26,78	28,43	24,45	30,31	29,15	19,63	28,73	17,32	14,29
	Actor 3b	16,16	16,80	14,73	15,38	13,88	16,76	14,41	12,58	15,50	10,51	9,30
	Unweighted 3,11	3,28	2,78	2,84	2,74	3,21	2,61	2,56	2,82	2,13	1,92

ACKNOWLEDGMENTS

This work was supported by the AgroParisTech Chair Sustainable Demand Supply Chain.


Tahar Z Boulmezaoud
email: tahar.boulmezaoud@univ-pau.fr

Mohammed El Rhabi
email: mohammed.elrhabi@math.jussieu.fr

A mortar spectral element method for 3D Maxwell's equations

Keywords: Maxwell's equations, mortar method, spectral elements, Galerkin approximations

In this paper we propose a mortar spectral element method for solving Maxwell's equations in 3D bounded cavities. The method is based on a non-conforming decomposition of the domain into the union of non-overlapping parallelepipeds. After proving an error estimate, we present some 3D computational results which confirm the performance of the method.

Introduction

The last two decades have seen a rapid development of numerical methods for solving Maxwell's equations in 2D and in 3D bounded regions of space. Let us recall e.g. the works of [START_REF] Hoppe | A penalty method for approximate solution of stationary Maxwell equations[END_REF][START_REF] Hoppe | Crank-Nicolsen-Galerkin approximation for Maxwell's equations[END_REF], Monk (1992aMonk ( ,b, 1993[START_REF] Monk | Finite Element Methods for Maxwell's Equations[END_REF], Beck et al. (1999), Beck & Hiptmair (1999), [START_REF] Hiptmair | Multigrid method for Maxwell's equations[END_REF], [START_REF] Costabel | A coercive bilinear form for Maxwell's equations[END_REF], [START_REF] Costabel | Singularities of Maxwell interface problems[END_REF], 2000), [START_REF] Buffa | Boundary elements methods for Maxwell transmission problems in Lipstchitz domains[END_REF], [START_REF] Toselli | Overlapping Schwarz methods for Maxwell's equations in three dimensions[END_REF] and [START_REF] Toselli | An iterative substructuring method for Maxwell's equations in two dimensions[END_REF]. Among the existing methods, domain decomposition techniques turned out to be very efficient since they bring a serious flexibility in the discretization of the original problem.

Our goal in this paper is to use a mortar spectral element method for approximating time-dependent and time-harmonic Maxwell's equations in a 3D bounded cavity.

The mortar method was introduced in Bernardi et al. (1990a,b) for coupling spectral and finite element approximations. It has mainly two advantages. Firstly, it allows the coupling of different, independent and non-matching numerical discretizations inside the domain of interest. Secondly, it is naturally predisposed to parallel computation (see, e.g. [START_REF] Fisher | Parallel spectral element solution of the Stokes problem[END_REF].

Mortar method with finite elements, spectral elements or wavelets were successfully applied to second-order elliptic problems (see, e.g. Bernardi et al., 1990b;[START_REF] Wohlmuth | A mortar finite element method using dual spaces for the lagrange multiplier[END_REF][START_REF] Bertoluzza | The mortar method in the wavelet context[END_REF]. It was also applied to several other problems including fourth-order equations (see [START_REF] Belhachmi | Resolution of fourth order problems by the mortar element method[END_REF], Stokes equation (Débit & Maday, 1987) and 2D and 3D Maxwell's equations (with finite elements) (see [START_REF] Rapetti | Approximation des équations de la magnétodynamique en domaine tournant par la méthode des éléments avec joints[END_REF][START_REF] Ben Belgacem | The mortar finite element method for 3D Maxwell equations: first results[END_REF].

Let us describe the Maxwell's equations we deal with in this paper. Let Ω be a bounded open set of R 3 , with Lipschitz boundary ∂Ω and a unit outward normal n. Assume also that Ω is occupied by a heterogeneous, linear and isotropic material and that its boundary ∂Ω is a perfect conductor. If E(x, t) and H(x, t) denote, respectively, the electric and magnetic fields in Ω, then Maxwell's equations state that

µ∂ t H + curl E = 0, ∂ t E -curl H = -j, div (µH) = 0, div ( E) = ρ,
(1.1) with ∂ρ ∂t + div j = 0.

(1.

2)

The boundary conditions we consider here are of the form E × n = 0 on ∂Ω, µH • n = 0 on ∂Ω.

(1.3)

Here and below ρ and j denote, respectively, the charge and the electric current densities. The coefficient µ(x) is the magnetic permeability of the material in Ω and (x) is its electric permittivity. We suppose that (x) and µ(x) are real, bounded and positive functions depending on the space variable x, i.e. there exist four constants 0 , ∞ , µ 0 and µ ∞ such that 0 < 0 (x) ∞ and 0 < µ 0 µ(x) µ ∞ a.e. in Ω.

(1.4)

In a time-dependent regime, the evolution equations (1.1) are completed with the initial conditions E(x, t = 0) = E 0 (x), H(x, t = 0) = H 0 (x).

(1.5)

It is usual to rewrite system (1.1) in terms of one unknown, say the magnetic field H or the electric field E. Here, we choose the formulation in terms of H. Eliminating E from the equations above gives µ∂ 2 t H + curl( -1 curl H) = curl( -1 j), div (µH) = 0.

(1.6)

The magnetic field is subject to the boundary conditions µH • n = 0 on ∂Ω, -1 (curl Hj) × n = 0 on ∂Ω,

(1.7)

and to initial conditions (1.5). In a time-harmonic regime the magnetic field satisfies curl( -1 curl H) -ω 2 µH = curl( -1 j).

(1.8)

The space discretizations of (1.6) and (1.8) are often distinguished in the literature. Indeed, an implicit time-stepping of (1.6) leads usually to a system of the form (see, e.g. [START_REF] Hoppe | Crank-Nicolsen-Galerkin approximation for Maxwell's equations[END_REF] curl( -1 curl u) + τ µu = curl( -1 j) + u 0 in Ω, µu

• n = 0 o n ∂Ω, curl u × n = j × n on ∂Ω, (1.9)
where τ is a positive parameter. The weak formulation of this problem rests on the bilinear form ( -1 curl u, curl v) + (τ µu, v) which is coercive on H (curl; Ω).

In contrast, the time-harmonic equation (1.8), written in a weak form, involves the bilinear form ( -1 curl u, curl v)-ω 2 (µu, v) which is not coercive either on H (curl; Ω) or on H 0 (div ; µ, Ω)∩H (curl; Ω) even if ω > 0. Two techniques might be used to deal with this problem. One of these is to penalize the weak problem by adding the term (div (µu), div (µv)) (actually, this term is 0 when u = H) (see, e.g. [START_REF] Hoppe | A penalty method for approximate solution of stationary Maxwell equations[END_REF][START_REF] Costabel | A coercive bilinear form for Maxwell's equations[END_REF]. In this case, the bilinear form

( -1 curl u, curl v) + (div (µu), div (µv))
is coercive on the space H 0 (div u, µ, Ω) ∩ H (curl; Ω). Unfortunately, as will be noted in Remark 1 below, this penalty method employed with mortar spectral elements induces some complications which are still unresolved. The second technique, which we expose in Section 2.2, consists of finding the solution of (1.8) as the limit of a quickly converging sequence of solutions of positive definite problems of form (1.9). A considerable advantage of such a technique is the possibility of using in practice a common method for solving time-dependent and time-harmonic Maxwell's equations. Thus, we choose to use the second technique in this paper.

Let us notice that if Ω is a non-convex polyhedron, then the magnetic field can have some singularities which are due to reentrant corners and edges. The reader can refer to [START_REF] Costabel | Singularities of electromagnetic fields in polyhedral domains[END_REF] and references therein for a characterization of these singularities. In that case, it is well known that H 1 elements are not sufficient to approximate the singular part of the solution (see for instance [START_REF] Costabel | A coercive bilinear form for Maxwell's equations[END_REF][START_REF] Costabel | Singularities of electromagnetic fields in polyhedral domains[END_REF]. Similarly, singularities could appear also at interfaces if the parameters and µ are discontinuous (see, e.g. [START_REF] Costabel | Singularities of Maxwell interface problems[END_REF]. The analysis and the approximation of these singularities are beyond the scope of this paper. Here, we shall suppose that the solution we look for is piecewise smooth (see the subsequent Theorem 1), and we do not make for the moment any other assumption, either on the domain Ω or on the coefficients and µ. Of course, this smoothness assumption is plausible if Ω is convex or if it has a regular boundary and if the coefficients and µ are constant or globally smooth (see, e.g. Lemma 2.3 in Monk, 1992a). In other situations, the method we expose here could be used only for computing the smooth part of the solution. The singular part must be treated separately.

An outline of this paper is as follows. In Section 2, we recall and state briefly some definitions and properties of the function spaces used as a framework. Then, we prove that each solution of the time-harmonic Maxwell's equations can be obtained as the limit of a rapidly converging sequence of solutions of a coercive problem.

Section 3 is devoted to the presentation of a mortar method with spectral elements for Maxwell's equations and to the proof of its numerical convergence.

The purpose of Section 4 is twofold. Firstly, we show that in practice the implementation of the mortar spectral element could be simplified by means of a theoretical treatment of the transmission conditions. Secondly, we display some 3D computational results which confirm the convergence of the method in several cases.

Preliminaries

Notations

In the sequel, for any integer d

1, any open set O of R d (d = 1, 2, 3) and any positive integers N 1 , . . . , N d , P N 1 ,...,N d (O), stands for the space of all the polynomial functions on O whose degree with respect to the variable x i , 1 i d, is less than or equal to

N i . If N 1 = • • • = N d = N , then we write P N (O) instead of P N 1 ,...,N d (O).
As usual, L 2 (O) refers to the space of square integrable functions on O. This space is equipped with the inner product

(u, v) O = O u(x)v(x) dx,
and with the associated norm u 0,O = (u, u)

1/2 O . The subscript O will be dropped in these notations when the domain is Ω.

In what follows H s (O), s being real, will be the usual Sobolev space of order s and • s,O its norm. When O is an open set of R 3 we write

H (curl; O) = {v ∈ L 2 (O) 3 , curl v ∈ L 2 (O) 3 }.
This is a Hilbert space equipped with the inner product

(u, v) curl,O = (curl u, curl v) 0,O + (u, v) 0,O ,
and with the norm

u H (curl;O) = ( v 2 0,O + curl v 2 0,O ) 1/2 .
Recall that if O is bounded and has a Lipschitzian boundary, then the tangential trace operator u → u×n is well defined and is continuous from H (curl; O) into H -1/2 (∂O) 3 . Moreover, the following Green's formula holds

∀ v ∈ H (curl; O), ∀ w ∈ H 1 (O) 3 , (curl v, w) = (v, curl w) + v × n, w ∂O .
(2.1)

Let H 0 (curl; O) refer to the subspace of vector fields in H (curl; O) with a vanishing tangential component on ∂O. Finally, we introduce the space

H (div ; µ, Ω) = {v ∈ L 2 (Ω) 3 , div (µv) ∈ L 2 (Ω)},
equipped with the norm u H (div ;µ,Ω) = ( v 2 0,Ω + div (µv) 2 0,Ω ) 1/2 , and its subspace

X T (µ; Ω) = {u ∈ H (curl; Ω) ∩ H (div ; µ, Ω) | µu • n = 0 on ∂Ω} , endowed with the norm u X T (µ;Ω) = { u 2 0,Ω + curl u 2 0,Ω + div (µu) 2 0,Ω } 1/2 . (2.2)
The following semi-norm on X T (µ; Ω) will be useful subsequently

|u| X T (µ;Ω) = { curl u 2 0,Ω + div (µu) 2 0,Ω } 1/2 . (2.3)
Actually, this semi-norm is a norm on X T (µ; Ω) equivalent to the norm (2.2) (see [START_REF] Monk | Finite Element Methods for Maxwell's Equations[END_REF][START_REF] Costabel | A coercive bilinear form for Maxwell's equations[END_REF], see also El Rhabi, 2002, for a detailed proof). Consequently, we set

ω 0 = inf v∈X T (µ;Ω);div (µv)=0; v =0 -1/2 curl v 0,Ω µ 1/2 v 0,Ω . (2.4)
This constant is strictly positive and depends only on Ω and on the parameters and µ.

A relaxation method for time-harmonic Maxwell's equations

As mentioned in the introduction, the solution of the time-harmonic problem (2.5) can be obtained as the limit of a sequence of solutions of problems of the form (1.9). The main goal of such a procedure is to prove that the numerical solution of both time-dependent and time-harmonic Maxwell's systems can be obtained in practice by solving a positive definite elliptic problem. In this preliminary section, we construct such a sequence and we prove its convergence.

Let us consider the time-harmonic Maxwell's system

curl -1 curl u -ω 2 µu = curl( -1 j) in Ω, div (µu) = 0 i nΩ, -1 (curl u -j) × n = 0 on ∂Ω.
(2.5)

It is well known that this problem satisfies the Fredholm alternative (see [START_REF] Leis | A finite element method for approximating the time-harmonic Maxwell's equations[END_REF], see also [START_REF] Dautray | Analyse Mathématique et Calcul numérique[END_REF][START_REF] Dautray | Analyse Mathématique et Calcul numérique[END_REF][START_REF] Boulmezaoud | On time-harmonic Maxwell's equations in Lipschitz and multiplyconnected domains of R 3[END_REF][START_REF] Boulmezaoud | On time-harmonic Maxwell's equations in Lipschitz and multiplyconnected domains of R 3[END_REF]. Now, let us consider the following iteration procedure; given u 0 = 0, for any k 0 find u k+1 as the solution of the weak problem

∀ v ∈ X T (µ; Ω)( -1 curl u k+1 , curl v) + τ (µu k+1 , v) = (ω 2 + τ )(u k , µv) + ( -1 j, curl v), (2.6)
where τ is a fixed real parameter not depending on k and which will be chosen to ensure convergence. Observe that the bilinear form on the left-hand side of (2.6) is coercive if τ > -ω 2 0 . It follows from the Lax-Milgram theorem that the sequence (u k ) k 0 is well defined. The convergence of that sequence is stated as follows.

PROPOSITION 1 Suppose that ω < ω 0 and τ > -

ω 2 + ω 2 0 2 . (2.7)
Then, the sequence (u k ) k 0 converges strongly in X T (µ; Ω) to the unique solution of (2.5) in X T (µ; Ω). Furthermore, one has

u k -u X T (µ;Ω) 1 2 ∞ (ω 2 0 + |τ |) 1 2 κ k+1 u Y T (Ω,µ, ) ,
where

κ = |ω 2 + τ | ω 2 0 + τ < 1.
Proof. Consider the sequence Φ k = u ku, where u ∈ X T (µ; Ω) is the solution of (2.5). Then, div (µΦ k ) = 0 and

( -1 curl Φ k+1 , curl v) + τ (Φ k+1 , µv) = (ω 2 + τ )(Φ k , µv), (2.8)
for all v ∈ X T (µ; Ω). It follows that

(ω 2 0 + τ ) √ µΦ k+1 0,Ω |ω 2 + τ | √ µΦ k 0,Ω . Thus, if τ > -ω 2 0 one gets √ µΦ k+1 0,Ω κ √ µΦ k 0,Ω .
Hence, √ µΦ k+1 0,Ω κ k+1 √ µu 0,Ω .

Moreover, from (2.8) then

-1 ∞ |Φ k+1 | 2 X T (µ;Ω) (-inf(τ, 0) √ µΦ k+1 0,Ω + |ω 2 + τ | √ µΦ n 0,Ω ) √ µΦ k+1 0,Ω [-inf(τ, 0)κ k+1 + |ω 2 + τ |κ k ]κ k+1 √ µu 2 0,Ω (ω 2 0 + |τ |)κ 2k+2 √ µu 2 0,Ω .
If ω < ω 0 and 2τ > -(ω 2 0 + ω 2 ), then κ < 1. In that case the sequence {u k } converges strongly to u in X T (µ; Ω).

Of course, the definition of κ in Proposition 1 suggests that the best choice for τ would be τ = -ω 2 < 0. With this choice, one is back to the original problem and the relaxation procedure is useless. However, in order to get a positive definite bilinear form in the weak formulation, it is better to choose τ > 0 in which case one is led at each iteration to solving a coercive problem.

In the remainder of this paper we deal only with problem (1.9).

The mortar method

The discrete space

In what follows, for each integer n 0, L n (x) stands for the Legendre polynomial of one variable and of degree equal to n. Recall that Legendre polynomials form an orthogonal system in L 2 (] -1, 1[) and satisfy the following basic properties:

L n (1) = 1, 1 -1 L n (x) 2 dx = 2 2n + 1 , 1 -1 L n (x) 2 dx = n(n + 1), (3.1)
for each integer n 0. Given an integer N 2, the Gauss-Lobatto grid on ] -1, 1[ of order N is formed by the zeros [START_REF] Davis | Methods of Numerical Integration[END_REF]Szegö, 1978;[START_REF] Bernardi | Approximation Spectrale de Problèmes aux Limites Elliptiques[END_REF]. Moreover, there exist positive weights ρ N ,0 , . . . , ρ N ,N such that

ξ N ,0 = -1 < ξ N ,1 < • • • < ξ N ,N = 1 of the polynomial (1 -x 2 )L N (x) (see, e.g.
∀ p ∈ P 2N -1 (] -1, 1[), 1 -1 p(x) dx = N i=0 ρ N ,i p(ξ N ,i ).
(3.2) Suppose now that Ω has a non-overlapping partition of the form

Ω = K k=1 Ω k ,
where Ω 1 , . . . , Ω K are K disjoint axiparallel rectangular parallelepipeds of R 3 . Assume that the intersection of two sub-domains is either empty or a point or a segment or a rectangle.

The skeleton S of this decomposition is defined by A 2D illustration of an excluded domain decomposition since some mortars cannot be chosen as the whole edges of sub-domains (right).

S = ⎛ ⎝ K k=1 K j=1 ∂Ω k ∩ ∂Ω j ⎞ ⎠ .
Suppose that this skeleton admits a partition of the form

S = m s=1 F - s ,
where, for each s m, F - s is a whole face of Ω k(s) for some k(s) K . Let us underline that this last assumption is not restrictive since it allows conforming and non-conforming geometrical decompositions (see 2D illustrations in Fig. 1). However, it excludes some domain decompositions for which some of the faces F - s could not be chosen as complete faces of sub-domains (see the 2D illustration in Fig. 1).

The faces F - s , s = 1, . . . , m, are called mortar faces. In the sequel, given an integer s m, we write

E(s) = { K | ∂Ω ∩ F - s = ∅}, E(s) = E(s) -{k(s)}.
We call the slave side of F - s the face F - s itself considered as a part of the boundary of Ω k(s) . By its master side we mean F - s considered as a part of the boundary of ∈ E(s) Ω . If u is a function on Ω, then u + refers to its value on the master side of the skeleton, while u -refers to its value on the slave side. The jump of u across the skeleton is given by [u] = u +u -. Consequently, the restriction of any vector field in

H (curl; Ω) to a sub-domain Ω k is in H (curl; Ω k ) and satisfies the transmission condition [u × n k ] = 0 (3.3) at each interface ∂Ω k ∩ ∂Ω . Notice that for almost every x ∈ F - s there exists a unique ∈ E(s) such that x ∈ ∂Ω k(s) ∩ ∂Ω . Now, consider K integers N 1 , . . . , N K such that N k 2 for all k K and define the global discretization parameter δ = (N 1 , . . . , N K ) ∈ N K . (3.4)
For any positive integer j N min , δj will be the multi-index (N 1j, . . . , N Kj). We set

N max = max 1 k K N k , N min = min 1 k K N k , M s = N k(s) , η δ = max 1 s m max ∈E(s) N min ∈E(s) N N max N min .
FIG.

2. An example of the mortar decomposition in a 3D configuration.

Observe that

∀ s m, ∀ ∈ E(s), η -1 δ N M s η δ N . Now, to each sub-domain Ω k , k = 1, . . . , K , we associate a Gauss-Lobatto grid Ξ k = {ξ k α = (ξ k N k ,i , ξ k N k , j , ξ k N k , ); α = (i, j, ) with |α| = sup(i, j, ) N k },
obtained by translation and homothety in each direction of the nodes

ξ N k ,0 , ξ N k ,1 , . . . , ξ N k ,N k . The corresponding weights ρ (k) α , |α| N k , are given by ρ (k) α = ρ N k ,i ρ N k , j ρ N k , |Ω k | 8 , with α = (i, j, ).
Define also the discrete product on the space of continuous functions on Ω k by

(u, v) Ω k ,N k = |α| N k ρ (k) α u(ξ k α )v(ξ k α ).
Thus, if u and v are such that

uv| Ω k ∈ P 2N k -1 (Ω k ), then (u, v) Ω k ,N k = Ω k u(x)v(x) dx. (3.5)
Similarly, for any continuous vector functions u and v, define the scalar product

(u, v) Ω k ,N k =
It is well known that there exists a constant C such that (see [START_REF] Canuto | Approximation results for orthogonal polynomials in Sobolev spaces[END_REF], see also Remark 13.3 in Bernardi & Maday, 1994)

∀ ϕ ∈ P N k (Ω k ), ϕ 2 0,Ω k (ϕ, ϕ) Ω k ,N k C ϕ 2 0,Ω k . (3.6) Now, for any multi-index η = (n 1 , . . . , n K ) ∈ N K define W η (Ω) as the space of all the functions v ∈ L 2 (Ω) satisfying v |Ω k ∈ P n k (Ω k ).
The global Lagrange interpolation operator I δ associated with the grids

Ξ k , k = 1, . . . , K , is defined as follows: given a function v, continuous in each Ω k , I δ v is the only element of W δ (Ω) satisfying ∀ k K and ∀ ξ ∈ Ξ k , I δ v(ξ) = v(ξ).
The following estimate holds (see Bernardi & Maday, 1994, Theorem 14.2):

v -I δ v L 2 (Ω) c K k=1 N -s k k v H s k (Ω k ) , (3.7) and is valid for all v ∈ L 2 (Ω) such that for each k K , v| Ω k ∈ H s k (Ω k ) for some s k > 3/2. Now,
we are in a position to construct the space H δ (curl; Ω), the discrete counterpart of H (curl; Ω). Namely, H δ (curl; Ω) is defined as the space of all the vector functions v ∈ W δ (Ω) 3 satisfying

F - s (v + × n s -v -× n s ) • ψ dλ 2 = 0 ∀ ψ ∈ M s δ (F - s ), (3.8)
at each slave face F - s , s m. Here M s δ (F - s ) denotes a suitable space of Lagrange multipliers. More precisely, we choose

∀ 1 s m, M s δ (F - s ) = P N k (s)-2 (F - s ) 2 .
(3.9)

Another adequate choice could be the following

∀ 1 s m, M s δ (F - s ) = P N k(s) -2,N k(s) (F - s ) × P N k(s) ,N k(s) -2 (F - s ).
(3.10)

The forthcoming analysis remains valid for both the choices. Notice that the condition (3.8) is a weaker version of the transmission condition (3.3). Consequently, the inclusion H δ (curl; Ω) →H (curl; Ω) does not hold in general. Finally, define the broken norm

• H * (curl,Ω) as follows v 2 H * (curl,Ω) = K k=1 {(curl v, curl v) Ω k + (v, v) Ω k }.

The discrete problem

Here we wish to approximate the magnetic field H that satisfies the second-order equation

curl( -1 curl u) + τ µu = curl( -1 j) + u 0 in Ω, curl u × n = j × n on ∂Ω, (3.11)
where τ > 0 is a given parameter, j is the imposed current source and u 0 is a data field which comes from time-stepping in the case of time-dependent equations, or from the iteration procedure in the case of time-harmonic regime. This problem can be written in the weak form: find u ∈ H (curl; Ω) such that

a(u, v) = (v), (3.12)
where

a(u, v) = K k=1 Ω k -1 curl u • curl v dx + τ Ω k µu • v dx , (v) = K k=1 Ω k -1 j • curl v dx + Ω k u 0 • v dx .
The corresponding discrete problem is to find

u δ ∈ H δ (curl; Ω) such that a δ (u δ , v δ ) = δ (v δ ) ∀ v ∈ H δ (curl; Ω), (3.13) with a δ (u δ , v δ ) = K k=1 ( -1 curl u δ , curl v δ ) Ω k ,N k + τ (µu δ , v δ ) Ω k ,N k and δ (v δ ) = K m=1 ( -1 j, curl v δ ) Ω k ,N k + K m=1 (u 0 , v δ ) Ω k ,N k .
Here we supposed that -1 j and u 0 are continuous in each Ω k . Observe that there exist two constants α * (τ, µ 0 , ∞ ) and β * (τ, µ 0 , ∞ ), independent of δ, such that

∀ u δ ∈ H δ (curl; Ω), a δ (u δ , u δ ) α * u δ 2 H * (curl,Ω) , ∀ u δ , v δ ∈ H δ (curl; Ω), |a δ (u δ , v δ )| β * u δ H * (curl,Ω) v δ H * (curl,Ω) ,
(3.14) thanks to (1.4) and (3.6).

REMARK 1 In the case of time-harmonic equations, one could be tempted to consider directly the penalized discrete problem: find u ∈ X T,δ (µ; Ω) such that

a 2,δ (u, v) = 2,δ (v), (3.15)
where Here X T,δ (µ; Ω) denotes the discrete space corresponding to X T (µ; Ω) which is constructed similarly to H δ (curl; Ω) by adding a matching condition on the normal component at the mortars. With such an approach, a natural question arises, whether the semi-norm induced by the semi-inner-product

a 2,δ (u, v) = K k=1 {( -1 curl u, curl v) Ω k ,N k + (div (µu), div (µv)) Ω k ,N k -ω 2 (u, v) Ω k ,N k } and 2,δ (v) = K k=1 ( -1 j, curl v) Ω k ,N k .
K k=1 Ω k -1 curl u • curl v dx + Ω k div (µu) • div (µv) dx
may be a uniformly elliptic norm on the discrete space X T,δ (µ; Ω) (as is the semi-norm (2.3) in the continuous case). The difficulty of this unresolved question and the uncertainty about its affirmative answer are circumvented here by the use of the iterative algorithm proposed in Section 2.2.

Error estimate

The main theoretical result of this paper is the following.

THEOREM 1 Let u ∈ H (curl; Ω) be the solution of (3.12) and u δ the solution of (3.13). We set

λ = -1 (curl u -j). (3.16) Suppose that, for each k ∈ {1, . . . , K }, u ∈ H s k +1 (Ω k ) 3 , λ ∈ H σ k +1 (Ω k ) 3
for some real numbers s k > 1/2, σ k > 1/2. Suppose also that there exists a uniform constant η 0 such that η δ η 0 .

(3.17)

Then

u -u δ H * (curl,Ω) C K k=1 N 1/4-s k k u H δ k +1 (Ω k ) + (log N k ) 1/2 N -σ k k λ H σ k +1 (Ω k ) . (3.18)
REMARK 2 The condition (3.17) can be dropped provided that the right-hand side of (3.18) is replaced by

C (η 1/4 δ + η 3/4 δ ) K k=1 N 1/4-s k k u H δ k +1 (Ω k ) + K k=1 η σ k δ (log N k ) 1/2 N -σ k k λ H σ k +1 (Ω k ) .
The proof is based on the following lemma.

LEMMA 1 We have

u -u δ H * (curl,Ω) C u -I δ u H * (curl,Ω) + inf v δ ∈H δ (curl;Ω) u -v δ H * (curl,Ω) + sup w δ ∈H δ (curl;Ω),w =0 b(u, w δ ) -b δ (u, w δ ) w δ H * (curl,Ω) + sup w δ ∈H δ (curl;Ω),w =0 a(u, w δ ) -(w δ ) w δ H * (curl,Ω) , (3.19) where b(u, w δ ) = a(u, w δ ) -(w δ ), b δ (u, w δ ) = a δ (I δ u, w δ ) -δ (w δ ).
Note that the last three terms on the right-hand side correspond, respectively, to the best approximation error, the consistency error and the numerical integration error.

Proof. This lemma is a slight modification of the standard Berger-Scott-Strang lemma (see [START_REF] Berger | Approximate boundary conditions in the finite element method[END_REF]. Indeed, for any v δ ∈ H δ (curl; Ω) one has

α * u δ -v δ 2 H * (curl,Ω) a δ (u δ -v δ , u δ -v δ ) = δ (u δ -v δ ) -a δ (v δ , u δ -v δ ) δ (u δ -v δ ) + a δ (I δ u -v δ , u δ -v δ ) -a δ (I δ u, u δ -v δ ) + a(u, u δ -v δ ) -a(u, u δ -v δ ) -(u δ -v δ ) + (u δ -v δ ).
Hence,

α * u δ -v δ H * (curl,Ω) β * I δ u -v δ H * (curl,Ω) + sup w δ ∈H δ (curl;Ω), w δ =0 |b(u, w δ ) -b δ (u, w δ )| w δ H * (curl,Ω) + sup w δ ∈H δ (curl;Ω), w δ =0 |a(u, w δ ) -(w δ )| w δ H * (curl,Ω) .
Finally, using the inequalities

u -u δ H * (curl,Ω) u -v δ H * (curl,Ω) + u δ -v δ H * (curl,Ω) , I δ u -v δ H * (curl,Ω) u -I δ u H * (curl,Ω) + u -v δ H * (curl,Ω) ,
completes the proof of (3.19). The rest of this section is devoted to the analysis of each term on the right-hand side of (3.18).

3.3.1

The interpolation and the best approximation error. The estimate of the best approximation error consists of finding a good interpolant of u in the discrete space H δ (curl; Ω). The construction of such an approximation constitutes one of the main difficulties in the numerical analysis of the mortar method.

PROPOSITION 2 Suppose that

u| Ω k ∈ H s k +1 (Ω k )
for some real numbers s k > 3/2 (k = 1, . . . , K ). Suppose also that the assumption (3.17) is satisfied. Then

u -I δ u H * (curl,Ω) + inf v δ ∈H δ (curl;Ω) u -v δ H * (curl,Ω) c K k=1 N 1/4-s k k u H s k +1 (Ω k ) , (3.20)
where c is not depending either on δ or on u.

The proof of Proposition 2 is divided into several steps. Let us start by introducing some useful notations. Let Λ =] -1, 1[ 3 be the unit cube and F =] -1, 1[ 2 ×{1} its upper face. Let e 1 = (1, 0, 0), e 2 = (0, 1, 0) and e 3 = (0, 0, 1) be the canonical basis of R rectangles G 1 , . . . , G r such that

• G i ∩ G j = ∅ for any i, j such that i = j, • F = r i=1 G i .
• The edges of each G k are parallel to the co-ordinate axes.

Denote by H 1 (F) the space of all the functions u ∈ L 2 (F) such that

u |G i ∈ H 1 (G i ) for all i ∈ {1, . . . , r }.
Step 1: some discrete operators For any integers N and M, π N ,M stands for the L 2 projector from L 2 (F) onto P N ,M (F). We denote by P * ,x N ,M (F) (resp. P * ,y N ,M (F)) the space of all the polynomial functions v ∈ P N ,M (F) satisfying v = 0 at x = ±1 (resp. at y = ±1). Define also the operator π * ,x N ,M as follows: for all u ∈ L 2 (Ω), π * ,x N ,M u is the unique element of P * ,x N ,M (F) satisfying (3.22)

F (u -π * ,x N ,M u) • ψ dx dy = 0, for any ψ ∈ P N -2,M (
We state the following result.

PROPOSITION 3 Let N 1 and M 1 be two integers. There exist two constants c 1 , c 2 , depending neither on N nor on M, such that

π * ,x N ,M u L 2 (F) c 1 N -1 r k=1 u H 1 (G k ) + N u L 2 (G k ) , (3.23) ∂(π * ,x N ,M u) ∂ x L 2 (F) c 2 N 1/2 r k=1 u H 1 (G k ) + N u L 2 (G k ) , (3.24)
for each u ∈ H 1 (F). Similar estimates hold for the operator π * ,y N ,M with N replaced by M and x replaced by y.

Proof. The following lemma is a first step towards the proof of Proposition 3.

LEMMA 2 Let N 1 and M 1 be two integers. There exists a constant c not depending either on N or on M such that for any function u ∈ H 1 (F) one has

(π N ,M u)(±1, •) L 2 (]-1,1[) c 1 N -1/2 r k=1 ( u H 1 (G k ) + N u L 2 (G k ) ).
(3.25) Proof. Let u ∈ H 1 (F). We set w = π N ,M u ∈ P N ,M (F) and

q N (x, y) = (L N (x) + L N +1 (x))w(1, y) ∈ P N +1,M (F).
We have on the one hand

1 -1 w(x, y) ∂q N ∂ x (x, y) dx = 2w(1, y) 2 ,
since q N is orthogonal to the space P N -1,M (F). On the other hand, we can write

F w(x, y) ∂q N ∂ x (x, y) dx dy = F u(x, y) ∂q N ∂ x (x, y) dx dy. Hence, 2 1 -1 w 2 (1, y) dy = r k=1 G k u(x, y) ∂q N ∂ x (x, y) dx dy = r k=1 - G k ∂u ∂ x (x, y)q N (x, y) dx dy + ∂G k u(x, y)q N (x, y)n • e 1 dσ c r k=1 N -1/2 ∂u ∂ x L 2 (G k ) + u L 2 (∂ G k ) w(1, •) L 2 (]-1,1[) ,
where we used the Cauchy-Schwarz inequality combined with the estimates

q N 2 L 2 (F) = L N 2 L 2 (]-1,1[) + L N +1 2 L 2 (]-1,1[) w(1, •) 2 L 2 (]-1,1[) C N -1 w(1, •) 2 L 2 (]-1,1[) |q(x, y)| max -1 x 1 |L N -1 (x)| + max -1 x 1 |L N -2 (x)| |w(1, y)| 2|w(1, y)| ∀(x, y) ∈ F.
Recall that there exists a constant C such that for any > 0 the inequality

v L 2 (∂ G k ) c( v H 1 (G k ) + -1 v L 2 (G k ) )
holds for any v ∈ H 1 (G k ). Taking v = u and = N -1/2 gives (3.25). We need also the following lemma.

LEMMA 3 There exists a constant c such that for any integers N and M and any polynomial function p in P N ,M (] -1, 1[ 2 ), the following inequality holds

p L 2 (T ) C N p L 2 (]-1,1[ 2 ) , (3.26)
where

T = ({-1}×] -1, 1[) ∪ ({1}×] -1, 1[).
Proof. Let p be in P N ,M (] -1, 1[ 2 ). We write its expansion in the basis of Legendre polynomials

p(x, y) = N k=0 M j=0 a k, j L k (x)L j (y),
p 2 L 2 (]-1,1[ 2 ) = N k=0 M j=0 4a 2 k, j
(2k + 1)(2 j + 1) .

On the other hand, we have

p 2 L 2 (T ) = M j=0 N k=0 (-1) k a k, j L j (•) 2 L 2 (]-1,1[) + M j=0 N k=0 a k, j L j (•) 2 L 2 (]-1,1[) = M j=0 2 N k=0 (-1) k a k, j 2 2 j + 1 + M j=0 2 N k=0 a k, j 2 2 j + 1 .
Then, using the Cauchy-Schwarz inequality leads to

p 2 L 2 (T ) = 4 N k=0 (2k + 1) M j=0 N k=0 a 2 k, j (2k + 1)(2 j + 1) C N 2 p 2 L 2 (]-1,1[ 2 )
. This completes the proof of Lemma 3. Now, we are in position to prove Proposition 3. Let u ∈ H 1 (F) and set w = π N -2,M u ∈ P N ,M (F) and v = π * ,x N ,M u. We prove easily that

v(x, y) = w(x, y) - 1 2 (L N -1 (x) + L N (x))w(1, y) - 1 2 (L N -1 (-x) + L N (-x))w(-1, y).
Thus,

v L 2 (F) w L 2 (F) + C N 1/2 w(1, •) L 2 (]-1,1[) + w(-1, •) L 2 (]-1,1[) .

Using (3.25) yields (3.23).

On the other hand, one has

F ∂v ∂ x 2 dx dy = - F v ∂ 2 v ∂ x 2 dx dy = - F u ∂ 2 v ∂ x 2 dx dy = r k=1 G k ∂u ∂ x ∂v ∂ x dx dy - ∂G k u ∂v ∂ x n • e x dσ r k=1 ( u H 1 (G k ) + N u L 2 (∂ G k ) ) ∂v ∂ x L 2 (G k ) ,
where we used the inverse inequality

∂v ∂ x n • e x L 2 (∂ G k ) C N ∂v ∂ x L 2 (G k ) ,
which follows from Lemma 3. This ends the proof of Proposition 3. Step 2: a lifting operator LEMMA 4 Let N 1 and M 1 be two integers. Then, for all v ∈ P * ,x N ,M (F) there exists a function

ψ ≡ R y N ,M v ∈ P N ,M,N (Λ) such that ψ = v, on F, ψe 2 × n = 0, on ∂Λ -F.
(3.27)

Moreover, there exists a constant c not depending either on N or on M such that

ψe 2 2 H (curl;Λ) c v L 2 (F) ∂v ∂ x L 2 (F) .
(3.28)

Proof. In the following, c and C denote two generic constants not depending on N and M. We set

ψ(x, y, z) = ϕ N (z)v(x, y),
where

ϕ N ∈ P N (] -1, 1[) is a function which satisfies ϕ N (1) = 1, ϕ N (-1) = 0,
and which will be chosen later. It is quite clear that ψ fulfills (3.27). On the other hand, we have

ψ 2 L 2 (Λ) = ϕ N 2 L 2 (]-1,1[) v 2 L 2 (F) , curl(ψe 2 ) 2 L 2 (Λ) = ϕ N 2 L 2 (]-1,1[) ∂v ∂ x 2 L 2 (F) + v 2 L 2 (F) ϕ N 2 L 2 (]-1,1[) .
(3.29)

Thus, if v = 0 curl(ψe 2 ) 2 L 2 (Λ) = ϕ N 2 L 2 (]-1,1[) ω 2 v + ϕ N 2 L 2 (]-1,1[) v 2 L 2 (F) ,
where where for any m ∈ {N , N + 1}, p m (z) is defined as

ω v = ∂v ∂ x L 2 (F) v L 2 (F) . Note that c ω v C N 2 , (3.30) thanks to the inverse inequality ∂v ∂ x L 2 (F) C N 2 v L 2 (F) , (
p m (z) = k 0 1 ω 2(k+1) v L (2k+1) m (z),
while α and β are given by

α = 1 2 p N (1) , β = 1 2 p N +1 (1)
.

Observe that

|α| cω v , |β| cω v , since p m (1) L m (1) ω 2 v cm 2 ω 2 v c ω v ,
for m = N or m = N + 1 (thanks to (3.30)). Observe also that ϕ N satisfies

ϕ N (1) = 1, ϕ N (-1) = 0 and ω 2 v ϕ N (z) -ϕ N (z) = αL N (z) + β L N +1 (z). (3.33)
Multiplying the latter by ϕ N (z) and integrating by parts on ] -1, 1[ gives

ω 2 v ϕ N 2 L 2 (]-1,1[) + ϕ N 2 L 2 (]-1,1[) = λ N , where λ N = ϕ N (1) + α + β.
On the other hand, multiplying (3.33) by 2(z + 1)ϕ N (z) and integrating by parts again yields

ω 2 v ϕ N 2 L 2 (]-1,1[) -ϕ N 2 L 2 (]-1,1[) = 2ω 2 v -2ϕ N (1) 2 -4(α + β)ϕ N (1) = 2ω 2 v -2λ 2 N + 2(α + β) 2 . Hence, 2ω 2 v ϕ N 2 L 2 (]-1,1[) = -2λ 2 N + λ N + 2ω 2 v + 2(α + β) 2 0 and 2 ϕ N 2 L 2 (]-1,1[) = 2λ 2 N + λ N -2ω 2 v -2(α + β) 2 0. It follows that -1 + 1 + 16ω 2 v + 16(α + β) 2 4 λ N 1 + 1 + 16ω 2 v + 16(α + β) 2 4 Cω v .
Finally, we get after substitution into (3.29)

curl(ψe 2 ) 2 L 2 (Λ) c ∂v ∂ x L 2 (F) v L 2 (F) , ψ 2 L 2 (Λ) c ω v v 2 L 2 (F) , v 3 L 2 (F) ∂v ∂ x L 2 (F) c v L 2 (F) ∂v ∂ x L 2 (F) ,
where we used Poincaré's inequality (3.32). Inequality (3.28) follows immediately. REMARK 3 The choice of the polynomial ϕ N is not as arbitrary as it seems. Indeed it minimizes the functional

ω 2 v ϕ N 2 L 2 (]-1,1[) + ϕ N 2 L 2 (]-1,1[) ,
on the set of all the polynomial functions p ∈ P N (] -1, 1[) satisfying the conditions p(1) = 1 and p(-1) = 0.

PROPOSITION 4 Let N 1 and M 1 be two integers. Then, there exists an operator R N ,M from L 2 (Λ)

3 onto P N ,M,M (Λ) × P N ,M,N (Λ) × {0} such that for all u ∈ L 2 (F) 3 , R N ,M u satisfies F (R N ,M u × n) • ψ dx dy = F (u × n) • ψ dx dy, for any ψ ∈ P N -2,M (F) × P N ,M-2 (F), R N ,M u × n = 0,
on ∂Λ -F.

(3.34) Moreover, there exists a constant c, independent of N and M, such that for all u

= (u 1 , u 2 , u 3 ) ∈ H (F) 2 × L 2 (F), we have R N ,M u H (curl;Λ) c M -1/4 r k=1 u 1 H 1 (G k ) + M u 1 L 2 (G k ) +N -1/4 r k=1 u 2 H 1 (G k ) + N u 2 L 2 (G k ) .
(3.35)

Proof. Let u = (u 1 , u 2 , u 3 ) ∈ L 2 (F) 3 and set R N ,M u = w 1 e 1 + w 2 e 2 ,
where

w 1 = (R x N ,M • π * ,y N ,M )u 1 and w 2 = (R y N ,M • π * ,x N ,M )u 2 .
Here R y N ,M denotes the lifting operator of Lemma 4, while R x N ,M is the operator obtained from R y N ,M by exchanging x and y. It is quite clear that R N ,M u satisfies (3.34). Furthermore, we have

R N ,M u 2 H (curl;Λ) c ⎛ ⎝ π * ,y N ,M u 1 L 2 (F) ∂(π * ,y N ,M u 1 ) ∂ y L 2 (F) + π * ,x N ,M u 2 L 2 (F) ∂(π * ,x N ,M u 2 ) ∂ x L 2 (F) ⎞ ⎠ c ⎛ ⎝ M -1/2 r k=1 u 1 H 1 (G k ) + M u 1 L 2 (G k ) 2 + N -1/2 r k=1 u 2 H 1 (G k ) + N u 2 L 2 (G k ) 2 ⎞ ⎠ ,
which is the desired result. Step 3: proof of Proposition 2

Let us now find an element u δ in the discrete space H δ (curl; Ω) which is a good interpolant of u. Firstly, we put

u δ = I δ u + w δ ,
where w δ will be chosen subsequently. Let z be the function defined on the skeleton S as follows

z = [I δ u -u] = [I δ u] on F - s , for s = 1, . . . , m.
Let F - s , s m, be a mortar face. For any ∈ E(s) we set Γ s, = F - s ∩∂Ω and we define the following extension of z

R s z = R M s z, in Ω k(s) , 0
R s z H * (curl,Ω) cM -1/4 s ∈ E(s) ( z × n s H 1 (F - s ∩∂Ω ) + M s z × n s L 2 (F - s ∩∂Ω ) ) cM -1/4 s ∈E(s) ( (I δ u -u) × n s H 1 (F - s ∩∂Ω ) + M s (I δ u -u) × n s L 2 (F - s ∩∂Ω ) ) cM -1/4 s ∈E(s) (N 1/2-s + M s N -1/2-s ) u × n H 1/2+s (∂Ω ) ) c(η 1/4 δ + η 3/4 δ ) K k=1 N 1/4-s k k u H s k +1 (Ω k ) .
Finally, we choose

w δ = m s=1 R s z.
Then,

u -u δ H * (curl,Ω) u -I δ u + m s=1 R s z H * (curl,Ω) C(η 1/4 δ + η 3/4 δ ) K k=1 N 1/4-s k k u H s k +1 (Ω k ) .
Using (3.17) ends the proof of (3.20).

3.3.2

The consistency error. Let λ be the vector field defined by (3.16). Note that λ ∈ H 0 (curl; Ω)

since curl λ = -τ µu + u 0 in Ω, λ × n = 0 on ∂Ω.
An estimate of the consistency error in terms of the vector field λ is given in the following.

PROPOSITION 5 Assume that for each k ∈ {1, . . . , K }, λ| Ω k ∈ H σ k +1 (Ω k ) for some real σ k > 1/2. Then sup w δ ∈H δ (curl;Ω),w δ =0 a(u, w δ ) -(w δ ) w δ H * (curl,Ω) c K k=0 η σ k δ (log N k ) 1/2 N -σ k k λ H σ k +1 (Ω k ) . (3.36) Proof. Observe that a(u, w δ ) -(w δ ) = K k=1 ( -1 curl u, curl w δ ) Ω k + τ (µu, w δ ) Ω k -( -1 j, curl w δ ) Ω k -(u 0 , w δ ) Ω k = K k=1 (λ, curl w δ ) Ω k -(curl λ, w δ ) Ω k .
Suppose now that λ ∈ H 1 (Ω k ) for any k K . Then, after an integration by parts we get

a(u, w δ ) -(w δ ) = K k=0 ∂Ω k (w δ × n) • λ dσ = m s=0 F - s [w + δ × n s -w - δ × n s ] • λ dσ = m s=0 F - s [w + δ × n s -w - δ × n s ] • (λ -ψ) dσ ,
where ψ is an arbitrary function defined on the skeleton S and is such that ψ ∈ P M s (F - s ) 3 for any s M. It follows that for any positive and sufficiently small parameters ρ k , k = 1, . . . , K , one has

|a(u, w δ ) -(w δ )| c m s=1 k∈E(s) ∂Ω k (w δ × n) • (λ -ψ) dσ c m s=1 k∈E(s) w δ × n H -1/2+ρ k (∂Ω k ) • (λ -ψ) H 1/2-ρ k (∂Ω k ) c m s=1 ⎛ ⎝ k∈E(s) ρ -1/2 k w δ × n H -1/2+ρ k (∂Ω k ) λ -ψ H 1/2-ρ k (F - s ) ⎞ ⎠ ,
where (λ -ψ) denotes the extension by zero of λ -ψ to ∂Ω k . The term ρ -1/2 k is due to the norm of this extension considered as a mapping from

H 1/2-ρ k (F - s ) into H 1/2-ρ k (∂Ω k ).
On the other hand, we have

w δ × n H -1/2+ρ k (∂Ω k ) w δ H ρ k (Ω k ) + curl w δ L 2 (Ω k ) ,
thanks to the formula (2.1). Then

w δ × n H -1/2+ρ k (∂Ω k ) cN 2ρ k k w δ H (curl;Ω k ) .
Finally, choosing

ρ k = (log N k ) -1 gives |a(u, w δ ) -(w δ )| c m s=1 ⎛ ⎝ k∈E(s) ρ -1/2 k N 2ρ k k M -σ k -ρ k s w δ H (curl;Ω k ) λ H σ k +1/2 (F - s ) ⎞ ⎠ c K k=0 ⎛ ⎝ k∈E(s) η σ k δ (log N k ) 1/2 N -σ k k w δ H (curl;Ω k ) λ H σ k +1/2 (F - s ) ⎞ ⎠ c K k=0 η σ k δ (log N k ) 1/2 N -σ k k λ H σ k +1 (Ω k ) w δ H δ (curl;Ω) .

3.3.3

The numerical integration error. Our task here is to give an estimate of the error due to the use of quadrature formulas for computing the integrals appearing in the discrete problem. We state the following.

LEMMA 5 Suppose that the vector field λ, defined by (3.16), satisfies the assumptions of Proposition 5.

Then, there exists a constant c not depending either on δ or on u such that sup

w δ ∈H δ (curl;Ω),w =0 b(u, w δ ) -b δ (u, w δ ) w δ H δ (curl;Ω) c K k=1 N -σ k k λ H σ k +1 (Ω k ) , (3.37) where b(u, w δ ) = a(u, w δ ) -(w δ ), b δ (u, w δ ) = a δ (u, w δ ) -δ (w δ ).
Proof. The estimate of the numerical integration is a classical exercise. In fact, consider two functions It follows that sup

z 1 ∈ W δ-1 (Ω) 3 , z 2 ∈ W δ-1 (Ω) 3 . One has |b(u, w δ ) -b δ (u, w δ )| K k=1 (λ -z 1 , curl w δ ) Ω k + (τ µu -u 0 + z 2 , w δ ) Ω k + K k=1 (z 1 -λ, curl w δ ) Ω k ,N k + (z 2 -τ µu + u 0 , w δ ) Ω k ,N k K k=1 (λ -z 1 , curl w δ ) Ω k + (z 2 -curl λ, w δ ) Ω k + K k=1 (z 1 -λ, curl w δ ) Ω k ,N k + (curl λ -z 2 , w δ ) Ω k ,N k .
w δ ∈H δ (curl;Ω), w δ =0 |b(u, w δ ) -b δ (u, w δ )| w δ H δ (curl;Ω) c λ -z 1 L 2 (Ω) + curl λ -z 2 L 2 (Ω) + I δ λ -z 1 L 2 (Ω) + I δ (curl λ) -z 2 L 2 (Ω) c λ -z 1 L 2 (Ω) + curl λ -z 2 L 2 (Ω) + I δ λ -λ L 2 (Ω) + I δ (curl λ) -curl λ L 2 (Ω) .
Choosing z 1 = I δ-1 λ, z 2 = I δ-1 λ and using (3.7) completes the proof of (3.37).

Numerical implementation

The goal of this section is twofold. Firstly, we give some details about the numerical implementation of the system (3.13). More precisely, we prove that the matching conditions (3.8) can be written explicitly without having to resort to numerical inversion of the resulting linear system. This calculus induces the construction of an explicit basis of the discrete space H δ (curl; Ω). Secondly, we present some computational results obtained with a 3D code. These results confirm the good convergence of the method and the relaxation algorithm exposed in Section 2.2.

A basis of H δ (curl; Ω). The treatment of the matching conditions

The definition of the space H δ (curl; Ω) introduced in Section 3.1 depends on the choice of the Lagrange multiplier spaces M s δ (F s ). Here we consider the spaces M s δ (F s ) defined by (3.9). The other choice given by (3.10) can be treated exactly by the same method.

Let ξ k , 1 k G, be the nodes of the grid Ξ . For each s ∈ {1, . . . , m} we set

Λ int = {k | ξ k ∈ S} (nodes outside the skeleton), Λ + s = {k | ξ k ∈ Ξ ∩ S
for some ∈ E(s)} (nodes on the master side of the skeleton), s) ∩ S} (nodes on the slave side of the skeleton),

Λ - s = {k | ξ k ∈ Ξ k(
Λ - s = {k ∈ Λ - s | ξ k ∈ ∂ F - s }, Λ - s = {k ∈ Λ - s | ξ k ∈ ∂ F - s }. Notice that card(Λ - s ) = (N k(s) + 1) 2 , card( Λ - s ) = (N k(s) -1) 2 = dim(P N k(s) -2 ).
The grid Ξ can be partitioned as follows

Ξ = {ξ k , k ∈ Λ int } ∪ m s=1 {ξ k , k ∈ Λ - s } ∪ {ξ k , k ∈ Λ + s } .
Now, consider an arbitrary node ξ k . There exists an integer j K and a multi-index α such that ξ k = ξ ( j) α . The basis function of W δ (Ω), associated with the node ξ k , is defined as

ϕ k (x) = ( j) α (x), if x ∈ Ω j , 0,
elsewhere, where

α is the Lagrange polynomial associated with the node ξ

( j)
α in the Gauss-Lobatto grid Ξ j . Observe that the jump [ϕ k ] across the skeleton S vanishes if k ∈ Λ int . Now, any vector function u δ ∈ W δ (Ω) 3 can be written in the form

u δ = k∈Λ int u int k ϕ k (x) + m s=1 ⎛ ⎝ k∈Λ + s ∪Λ - s (u nor k n s + u k )ϕ k (x) ⎞ ⎠ , (4.1)
where u k • n s = 0 for any k ∈ ∪ m s=1 (Λ + s ∪ Λ - s ). Moreover, u δ belongs to H δ (curl; Ω) if and only if its tangential components satisfy the matching conditions (3.8). These conditions can be interpreted as follows; at each face F - s of the skeleton S, the tangential degrees of freedom

u k , k ∈ ∪ m s=1 (Λ + s ∪ Λ - s ), satisfy ∀ s ∈ {1, . . . , m}, ∀ p ∈ P N s -2 (F - s ), k∈Λ + s ∪Λ - s F - s [ϕ k ] • p(x) dσ u k = 0.
Hence, for any s ∈ {1, . . . , m} and any p ∈

P N s -2 (F - s ) k∈ Λ - s F - s [ϕ k ] • p(x) dσ u k = - k∈Λ + s F - s [ϕ k ] • p(x) dσ u k - k∈ Λ - s F - s [ϕ k ] • p(x) dσ u k .
(4.2) Let U + s , U - s and U - s be the vectors containing the values u k when k describes Λ + s , Λ - s and Λ + s , respectively. The linear system (4.2) can be written in the form

Q - s U - s = Q + s U + s + Q - s U - s , (4.3)
where Q - s , Q + s and Q - s are known matrices. Let U be the vector containing all the values u int

k (k ∈ Λ int ), u nor k for k ∈ ∪ m s=1 (Λ - s ∪ Λ + s ) and u k for k ∈ ∪ m s=1 Λ - s ∪ Λ + s .
The vector field containing all the unknowns u δ at all nodes of the grid Ξ is denoted by U . Notice that U contains both the real unknowns already stored in U and the apparent unknowns, u m , m ∈ ∪ m s=1 Λ - s . The vectors U and U are linked by a relation of the form

U = QU,
where Q is a rectangular matrix the rows of which are mostly those of the identity except those coming from the matrices

( Q - s ) -1 Q + s and ( Q - s ) -1 Q - s .
Finally, the discrete problem (3.13) can be written in the form

Q AQU = B, (4.4)
where A is the block matrix corresponding to the bilinear form a δ (•, •) given by

A = ⎛ ⎜ ⎜ ⎜ ⎝ A 1 0 0 0 0 A 2 0 0 0 0 . . . 0 0 0 0 A K ⎞ ⎟ ⎟ ⎟ ⎠ .
Here A k , k = 1, . . . , K , is the stiffness matrix associated with the local bilinear form Since the final matrix A = Q AQ is positive definite and symmetric, one can use a usual iterative method, e.g. Conjugate Gradient, for solving the linear system (4.4).

a k (u, v) = ( -1 curl u, curl v) Ω k ,N k + τ (µu, v) Ω k ,N k .
Let us prove now that the matching linear systems (4.3) can be inverted manually without resort to numerical inversion. We provide by this way a basis of the space H δ (curl; Ω). We suppose for simplicity that F - s =] -1, 1[ 2 ×{0}. We can write

u δ = k∈Λ int u int k ϕ k + m s=1 ⎛ ⎝ k∈Λ + s ∪Λ - s u nor k ϕ k n s + k∈Λ + s ∪ Λ - s u k ϕ k ⎞ ⎠ . (4.5)
Here the basis functions ϕ k are defined by

ϕ k = ϕ k - j∈ Λ - s q k, j ϕ j , 1 s m, k ∈ Λ + s ∪ Λ - s ,
where the coefficients q k, j are chosen so that

F - s [ ϕ k ] • q dσ = 0, ∀ q ∈ P N s -2 (F - s ). (4.6)
It is worth noting that for any s m and k ∈ Λ + s we have

ϕ k = ϕ k ,
on the master side of F - s , -j∈ Λ - s q k, j ϕ j , on the slave side of F - s .

It follows that the vector V k = (q k,1 , . . . , q k,θ ), θ = card( Λ - s ), is a solution of a linear system of the form QV k = B k .

For any k ∈ Λ + s ∪ Λ - s , define on F - s the functions

ψ k = j∈ Λ - s q k, j ϕ - j , ψk = ϕ - k , if k ∈ Λ - s , π N s ϕ + k , if k ∈ Λ + k ,
where ϕ + k (resp. ϕ - k ) denotes the value of ϕ k on the master (resp. the slave) side of F - s and π N s is the orthogonal projector from L

2 (F - s ) onto P N s (F - s ). Notice that for each k ∈ Λ + s one has ψk = N s , j=1
(2 + 1)(2 j + 1) 4

F - s ϕ + k (x, y) • L (x)L j (y) dx dy L (x)L j (y) dx dy. (4.7)
The matching condition (4.6) can be rewritten in the form

ψ k -ψk ∈ P ⊥ N s -2 (F - s ),
where one gets easily to the formula

P ⊥ N s -2 (F - s ) is the orthogonal projector of P N s -2 (F - s ) in P N s (F - s ). Since P ⊥ N s -2 (F - s ) = p(x)L j (y) + q(y)L k (x), p ∈ P N s (] -1, 1[ x ), q ∈ P N s (] -1, 1[ y ), N s -1 j, k N s } ,
ψ k = ψk - 1 2 ψk (x, 1)(L N s (y) + L N s -1 (y)) - (-1) N s 2 ψk (x, -1)(L N s (y) -L N s -1 (y)) - 1 2 ψk (1, y)(L N s (x) + L N s -1 (x)) - (-1) N s 2 ψk (-1, y)(L N s (x) -L N s -1 (x)) - δ 2 L N s -1 (x) + β 2 L N s (x) L N s -1 (y) - γ 2 L N s -1 (x) + λ 2 L N s (x) L N s -1 (y), (4.8)
where

2λ = ψk (1, 1) + (-1) N s ( ψk (-1, 1) + ψk (1, -1)) + ψk (-1, -1), 2β = ψk (1, 1) + (-1) N s ( ψk (-1, 1) -ψk (1, -1)) -ψk (-1, -1), 2γ = ψk (1, 1) + (-1) N s ( ψk (-1, 1) + ψk (1, -1)) -ψk (-1, -1), 2δ = ψk (1, 1) + (-1) N s ( ψk (-1, 1) -ψk (1, -1)) + ψk (-1, -1).
Finally, observe that

∀ k ∈ Λ + s ∪ Λ - s , ∀ j ∈ Λ - s , q k, j = ψ k ξ k(s) α( j) ,
where ξ k(s) α( j) ∈ F - s is the node associated with the function ϕ j . This ends the inversion of (4.3) and the construction of a basis of H δ (curl; Ω).

Computational tests

Our purpose here is to show some numerical experiments obtained with a 3D code for solving the discrete system (3.13). The aim of these tests is to compare the computational results with the numerical analysis given above. The domain of computation is the cube Ω = ] -1, 1[ 3 decomposed into three sub-domains

Ω 1 =] -1, 1[ 2 × ] -1, 0[, Ω 2 =] -1, 0[ × ] -1, 1[ × ]0, 1[, Ω 3 =]0, 1[ × ] -1, 1[ × ]0, 1[.
This decomposition is clearly non-conforming. The mortar faces are

F 1 = ] -1, 0[×] -1, 1[×{0}, k(1) = 2, F 2 = ]0, 1[×] -1, 1[×{0}, k(2) = 3, F 3 = {0}×] -1, 1[×] -1, 0[, k(3) = 3.
Figure 3 shows a vertical section of the domain

Ω = Ω 1 ∪ Ω 2 ∪ Ω 3 .
The tests we make here are gathered into two parts. The first tests concern the approximation of the problem (3.11) by the mortar spectral method exposed above (Examples 1 and 2). The second tests are devoted to approximating the harmonic problem (2.5) by means of the relaxation scheme exposed in Section 2.2. The mortar method is used at each iteration of that scheme. EXAMPLE 1: THE HOMOGENEOUS CASE The analytical example we use is given by

u = µ -1 v, j = µ -1 curl v + τ w, where v(x, y, z) = ⎛ ⎝ f (x) f (y) f (z) 4 f (x) f (y) f (z) -5 f (x) f (y) f (z) ⎞ ⎠ , w = ⎛ ⎝ 3 f (x) f (y) f (z) -2 f (x) f (y) f (z) -f (x) f (y) f (z) ⎞ ⎠ (4.9)
and f (t) = (sinh 2) sinh t -(sinh 1) sinh(2t). Here we choose k = µ k = 1 for k ∈ {1, 2, 3}. In Fig. 4 the three components of the vector field u at the section y = 1/2 are displayed. The same components are shown in the same figure after a computation by the mortar method with a 10-8-8 grid. These figures show, on the one hand, the continuity of the tangential trace across the mortars which is ensured by the matching conditions (3.8). On the other hand, they display the continuity of the normal trace which is implicit to the original equation.

Figure 5 shows the space distribution of the error when a 10-8-6 discretization is used. One can observe that the interface between sub-domains 1 and 2 is quasi-transparent since the error in its neighbourhood is very small. However, the error at interfaces between the sub-domain 3 and the sub-domains 1 and 2 is more significant. This is not due to the mortar matching conditions but to the diffusion of errors coming from sub-domain 3 where the grid is rough. In Figs 6 and 7 the relative L2 error on curl u, on u and the L 2 norm of div u (the latter is divided by the L 2 norm of curl u) are displayed, respectively, versus a parameter N for 1.5N -1.5N -N and 1.5N -N -1.5N discretizations. The three components of this heterogeneous solution are displayed in Fig. 8 (on the left) at the section y = 1/2. The distribution of the error in each component is also displayed at the same section (on the right).

Figure 7 shows the behaviour of the H (curl) norm of the error and the L 2 norm of the divergence versus N when a 1.5N -1.5N -N discretization is used. The decay of these errors confirms the convergence of the method in the case of heterogenous media, provided that the solution is sufficiently smooth.

4.2.2

Tests on the convergence of the iterative scheme for a harmonic system. Our task here is to confirm numerically the convergence of the iterative procedure proposed in Section 2.2 for solving the FIG. 9. The rate of convergence u n+1u n X T (µ;Ω) / u n X T (µ;Ω) versus the number of iterations for several values of the parameter τ (ω is fixed: ω = 0.1π ) (left). The rate of convergence versus the number of iterations for several values of ω (τ is fixed: τ = 10) (right). A 7-5-5 discretization is used for all the tests. harmonic problem (2.5). In all the tests, the parameters and µ are supposed constant. The current density j is given by j = µ -1 curl v -ω 2 w, (4.10)

where v and w are given by (4.9). The corresponding solution of (2.5) is u = µ -1 v.

The discretization parameters are also fixed; N 1 = 7, N 2 = 5, N 3 = 5. Figure 9 shows the rate of convergence u n+1u n X T (µ;Ω) u n X T (µ;Ω) versus the iteration number n for several values of τ (with ω = 0.1π) and for several values of ω (τ = 10). It is quite clear that the smaller τ and ω are, the faster is the convergence. Furthermore, there exists a value ω δ ≈ 0.4π such that the algorithm does not converge if ω > ω δ . All these observations confirm the predictions of Proposition 1. It is worth noting that the error does not vary significantly with the parameter τ . However, this parameter has a significant influence on the spectral properties of the matrix A, Â = Q AQ. More precisely, for small (but positive) values of τ , the convergence of the CG algorithm used for inverting the linear system becomes slow. Nevertheless, in such a situation one can use preconditioners to get rapid convergence.

583FIG. 1

 1 FIG.1. A 2D example of a conforming decomposition (left). A 2D illustration of a non-conforming decomposition (centre). A 2D illustration of an excluded domain decomposition since some mortars cannot be chosen as the whole edges of sub-domains (right).

  3 . Consider a collection of non-trivial openDownloaded from https://academic.oup.com/imajna/article-abstract/25/3/577/709236 by Ecole Nationale des Ponts et Chaussées (ENPC) user on 30 May 2018

  us introduce the operator π * ,y N ,M defined as: for all u ∈ L 2 (Ω), π * ,y N ,M u is the unique element of P * ,y N ,M (F) such that F (u -π * ,y N ,M u) • ψ dx dy = 0, for any ψ ∈ P N ,M-2 (F).

FIG. 3

 3 FIG.3. A vertical section of the 3D domain used for the numerical tests with a 15-15-10 Gauss-Lobatto grid.

FIG. 4 .

 4 FIG.4. Example 1. The three components of the exact solution at the section y = 1/2 (left) and of the approximate one (right) at the section y = 1/2. The computation of the approximate solution is done with a 10-8-8 mortar method.

EXAMPLE 2 :

 2 FIG.7. Example 1. The relative L 2 norm of the divergence (in a logarithmic scale) versus N for a 3N /2-3N /2-N discretization (solid line) and a 3N /2-N -3N /2 discretization (dashed line) (left). Example 2. The relative H (curl) norm of the error (solid line) and the L 2 norm of the divergence (dashed line) (in a logarithmic scale) versus N for a 3N /2-3N /2-N discretization (right).

  , elsewhere, where R M s is the lifting operator in Ω k(s) constructed from the operator R M s ,M s of Proposition 4 by translation, homothety and rotation. It follows from Proposition 4 that

Downloaded from https://academic.oup.com/imajna/article-abstract/25/3/577/709236 by Ecole Nationale des Ponts et Chaussées (ENPC) user on 30 May 2018

i=1 (u i , v i ) Ω k ,N k .Downloaded from https://academic.oup.com/imajna/article-abstract/25/3/577/709236 by Ecole Nationale des Ponts et Chaussées (ENPC) user on 30 May 2018

= 1.0,

=

= 1.5, µ 1 = 1.0, µ 2 = 2.0, µ 3 = 0.5. Downloaded from https://academic.oup.com/imajna/article-abstract/25/3/577/709236 by Ecole Nationale des Ponts et Chaussées (ENPC) user on 30 May 2018


PhD Gaëlle Petit

G Yannou-Le Bris

G Trystram

Individual choices influence on value chain sustainability

Context of the case study: choosing among 12 feed scenarios Can those actors achieve a same vision of food value chain sustainability?

Results

Outlook

 Sustainability performance needs to be discussed among value chain actors. Choice negotiation.  Big differences are observed between indicators classification but still alignment seems achievable even if not necessarily identical to individual sustainability orientations.

 Pillars classification of eco> social > env vs. Today environmental assessment. Need to strengthen the needs of parallel performance measurement system (acv, CSR enterprise, VC performance).

 If a value chain performs better on non prioritized indicators, it does not come out here, whereas in multicriteria evaluation (only one strategy taken into account) it may emerge as the best. Group evaluation vs. individuals evaluation  Limit : evaluation system is highly dependent on the selected indicators.

 Nature of data to collect, handle, sharing to carrying out future studies.

Weighting results for sustainability pillars

Methodology

Number of participants agree on the comparison in pairs of environmental indicators

0 None of the players have the same opinion 2/2 Two actors have the same opinion, two others have the same opinion 2/0 Two actors have the same opinion, the others differ 3 Three actors have the same opinion 4 The four players have the same opinion Decision is complex process because different interests Individual sustainability choices vs. value chain sustainability choices? Spontaneously same sustainability nature, strategy, objectives, evolution choice? Influence of the actor's strategy on their choices?

This work was supported by the Chair Sustainable Demand Supply Chain of AgroParisTech.

Number of participants agree on the comparison in pairs of economical indicators

Number of participants agree on the comparison in pairs of social indicators

Results for each #scenario and according to actors preferences