In today's French gendarmerie, commanding positions are put to the test of new conditions of exercise, both on the technological and relational levels. The removal of barriers and the immediacy in communication go hand in hand with the rise of a new condition of the gendarme that transforms the contours of the hierarchical relationship. In this new context, chiefs are encouraged to downplay some of their traditional prerogatives in favour of a more concerted approach to decision-making. This research 1 , conducted between 2014 and 2018, has focused on understanding the issues at stake in such a context and on analysing this (r)evolution of command in the gendarmerie2 . It is divided into three parts: an interpretative framework of a leader's various roles (I); a global analysis of the changes in the condition of the gendarme (II); and finally, the identification of a shifting process in modern command (III).

I) The command ing position in the divisional gendarmerie: a role-based approach

The analysis of command that is suggested here is a role and activity-based approach. It essentially rests on observation work carried out in the divisional gendarmerie, in order to break down the daily workload of the chiefs. The activities associated with the commanding position were thus translated into a repertoire of 10 roles. These roles are like relational tools which -when used in accordance with the situation -allow access to resources and privileges otherwise inaccessible. This repertoire is divided into three categories: symbolisation roles, monitoring roles and mediation roles.

The first roles (symbolisation) are based on the leader's visibility, both inside and outside his or her organisation. They generally involve representational activities and are centered on the use of the gendarmerie's symbols and cultural codes. The second roles (monitoring) are those that allow the chief to control the formal processes and internal movements of the organisation. They are often associated with administrative work and equipment-related arbitrations, but also with decision-making and strategy implementation. Finally, the last ones (mediation) bring together the various relational activities of proximity that unfold in privileged and trustbased exchanges rather than in public representations or formal events. As they are barely visible and often not very constraining -and although they have a strong strategic interest -they generally require the leader to take the initiative to use them. Represents the institution and the chain of command in formal moments. Is the pillar of the internal ceremonial procedures, participates in public events, and ensures good relations with partner institutions.

Ceremonies, official events, public relations.

Manager

Defines the framework of the mission and produces the signs that guide the collective action. Sets objectives, disseminates assessments and talking points, delivers speeches, formulates the strategy.

Production of talking points, speeches, valorisation, reprimands.

Leader

Ensures operational command and leads the staff into action. Serves as a role model and an example.

Involvement in the action, supervision of an intervention team.

Monitoring Validator

Validates the administrative acts of the unit. Gives legitimacy to the interactions and the topics covered.

Handling of signature books, signatures, meetings, responses to requests.

Dispatcher

Distributes resources of all types in the area of responsibility. These resources may be permanent or temporary. Decides on the initial allocation and the prioritisation of replacements.

Assignment decisions, organization of supply and replacement of equipment.

Manager

Organizes the day-to-day operations of the unit. Divides the workload related to the operation strategy.

Division of workload and rest periods.

Censor

Ensures that orders are carried out, goals achieved and actions compliant. Assesses the quality of the command. Carries out professional assessments and sanctions professional misconducts.

Computer control, inspections, command investigations, assessments, rewards, disciplinary measures.

Mediation Protector

Supports staff and absorbs various pressures. Listens to staff and makes them feel safe.

Visits, recognition, responses to complaints and requests.

Solver

Solves subordinates' problems and organisational dysfunctions. Finds solutions to internal crises and mediates or arbitrates disputes.

Coordination, crisis management, animation of prevention networks, dialogue.

Representative

Represents the interests and expectations of the staff and the organisation. Speaks out on sensitive issues.

Mediation, negotiations, request forwarding. The Leader's Role Repertoire and associated activities.

II ) A new employment relationship oriented towards fair treatment

Within the gendarmerie, the military status traditionally implies a real asymmetry between the chief and his subordinates, notably through the subordination of all to the service requirements. This situation is considered to be a normal consequence of the job, in that it stems directly from the double responsibility of the collective: that of ensuring permanent control of the territory and maintaining a permanent capacity for deployment and action. In this "traditional" configuration, asymmetry results from the fact that the management of the unit's forces is essentially left to the discretion of the chief, which gives the latter a number of advantages and possibilities of action. The gendarme himself, as a subordinate, is then strongly dependent in his daily life on the degree of fairness and accuracy with which his immediate superior makes the arbitrations necessary to maintain operational capacity.

On this specific point, the contemporary transformations of command cannot be understood outside of the recent changes in the employment relationship within the gendarmerie. Following, in particular, the strikes of 1989 and 2001, the French gendarmerie has in fact launched over the last fifteen years a series of voluntary reforms intended to transform the relationship between the personnel and their work and its constraints. The military status has gradually changed from a principle of total submission to the service requirements to a more egalitarian logic, based on new individual rights. It was thus enriched by three new fundamental rights for personnel -the right to speak out, the right to well-being and the right to free time -which reoriented daily management and decision-making methods. From the institution's perspective, these new fundamental rights are respectively represented by three major directions for change -developing a militarised rationale of labour relations, recruiting psychological and social support professionals on permanent long-term positions, and finally, revising the rules governing the organisation of working time.

Through these changes, the modern condition of the gendarme has been partially freed from the service requirements, as it allows the military to demand the application of individual rights that can sometimes contradict -directly or indirectly -the need for flexibility traditionally imposed by the logic of capacity. In a broader context where barriers are being removed and information can bypass the official channels via new technological tools or new intermediaries, the value of the leader is then evaluated and assessed as much on his operational results as on his ability to satisfy his subordinates. The monitoring roles, which until then benefited from the possibilities offered by the logic of capacity and the military culture, have thus seen their prerogatives reduced by a number of rules and new management modes. On the other hand, the mediation roles have gained prominence and now offer a number of advantages by placing the leader at the center of new dialogue and information sharing logics. This change in the possibilities offered by the leader's role repertoire necessarily modifies the way in which command manifests in daily interactions. Sometimes experienced as a hindrance, sometimes as an opportunity, it shows a shift in the playing rules of the hierarchical relationship and calls for a number of adjustments. It therefore reveals differences in career paths, leadership styles, and the resources actually available to make this transition a success. However, it is still possible for people in command positions to adapt their stance in such a way as to embrace the transformations of the institution and take full advantage of them. Adopting this new stance -called the "supportive stance" -can then be seen as a solution, and turns a number of constraints into opportunities to assert a new form of authority, better adapted to contemporary issues. However, it faces a number of challenges that should not be overlooked.

III) The transition from authority to support: a challenge for command

In general, the transition to a logic of support implies an enrichment of the practices and possibilities related to the mediation roles, which is supposed to compensate for an impoverishment of the monitoring roles. When they appropriate this transition, the leaders observed take control of a new system of relations that serves their territorial strategy. The freeing of speech, psychosocial concerns and the transformations of working time are presented as a set of opportunities, in that they allow new resources and new possibilities of action. For example, freeing speech within the institution allows managers to rebuild professional relations on a new basis, more focused on dialogue, staff buy-in to decisions and the involvement of local expertise.

In many ways, the change in the playing rules seems to be an opportunity to reinforce and legitimise the exercise of a commanding style based on proximity, thus reviving in some respects the military ideal of the charismatic and benevolent leader. Such a development would then solve the problem of the two distances that characterise the divisional gendarmerie: the first, subjective, which stems from the hierarchical deference of the military world; the second, more objective, linked to the units and resources being scattered all over the territory. However, if you look closely, seizing this opportunity for closer work relationships depends on a subtle balance between the chief maintaining his independence -or even his exclusivity -in decision-making and being attentive to the needs of the field. In other words, the leader must be careful to preserve his essential prerogatives while adopting an open attitude to new relational models.

The difficulties that may arise from this situation are due to many factors, such as the leader's isolation from his own hierarchy, the absence or intermittence of advisory resources, etc. From this point of view, it seems that the current situation does not create the same opportunities at different levels and locations, favoring in particular the highest command positions. This inequality within the territorial network, when combined with situations of relational blockages, can therefore generate conditions that may create a feeling of insecurity or even downgrading in the exercise of command. This process of revamping professional relations, far from being a given, should therefore include a logic of staff support so as not to generate counter-productive tensions at the local level.

Conclusion

Because of its particular situation, the French gendarmerie today is suggesting an interesting revamping of the traditional command doctrine. This new expertise, which should not be confused with a kind of "military management", offers opportunities in that it combines military culture and concerted decision-making practices informed by practical problems. This transition, which is not based solely on the establishment of new operating standards, assumes that each command position finds, at its own level, a number of balanced solutions. Beyond the issue of style or psychology, which is too often put forward to account for the difficulties of command, it is therefore on the real possibilities of adaptation that future developments must focus. In this respect, the situation of the most operational and/or most disadvantaged levels will have to come under close scrutiny, so that the tensions at the origin of former labour discontent do not eventually shift to the command levels which are closest to the field. ALFANO, Nicolas. From authority to support : roles and uses of roles in command in the Divisional Gendarmerie. doctoral thesis, Public management and territorial governance institute of Aix-Marseille, under the supervision of Robert FOUCHET, defended on December 7 th 2018. The full text is accessible to public without restriction.

Translated by SLT Quentin ROUSSEL and the French Gendarmerie Officers Academy Language Department

FROM AUTHORITY TO SUPPORT : CONTEMPORARY DEVELOPMENTS IN COMMAND IN THE FRENCH GENDARMERIE By Nicolas ALFANO, Doctor of management science Category Role Description Examples of activities Symbolisation Official

										Issue 40
										May 2019
	1 This note is a synthesis of a doctoral thesis : ALFANO, Nicolas. De l'autorité au soutien : rôles et usages des rôles dans le commandement en
	Gendarmerie Départementale (From authority to support : roles and uses of roles in command in the divisional Gendarmerie). Doctoral thesis,
	Institut de management public et de gouvernance territoriale (Institute of Public Management and Territorial Governance) of Aix-Marseille,
	supervised	by	Robert	FOUCHET,	defended	on	7	December	2018.	Available	at :
	https://www.researchgate.net/publication/331354642_ALFANO_Nicolas_-_De_l'autorite_au_soutien_-_These_de_Doctorat_2018

This applies mainly to the divisional gendarmerie, where all the work was done. However, some of the conclusions drawn here can be transposed to other workplaces.

The content of this publication is to be considered as the author's own work and does not engage the responsibility of the CREOGN.


Dr Laura Ascone

Ph.D In Language

Discourse analysis for action against jihadist radicalization

Faced with the jihadist attacks that have struck the West in recent years, researchers and professionals have rallied to understand the dynamics that can lead an individual to adhere to the ideology promoted by Daesh. If several disciplines such as psychology 1 , sociology 2 and computer science 3 have found their legitimacy in action-research programs on the phenomenon of jihadist radicalization, language sciences seem to encounter some difficulties. Therefore this study is conducted in a context where the role of language has been recognized but not sufficiently examined. Our objective here is to show how the analysis of jihadist discourse can contribute to understanding the phenomenon of radicalization and to formulating a strategy for dealing with it. This is not a research in language sciences, and more specifically in discourse analysis, which would take jihadist discourse as a simple case study. On the contrary, we want to bring the knowledge and methodologies of language sciences to the study of a phenomenon of global interest that is challenging all disciplinary boundaries. This discursive characterization leads to criteria for identifying radicalization on the web and to recommendations for messages that would enable us to counter jihadist radicalization.

Jihadist discourse and institutional counter-discourse

Corpus and methodology

The two official magazines published online by Daesh have proven to be an important source for studying the rhetorical strategies adopted by the jihadist group to fascinate its supporters and terrorize its enemies. These are the two magazines distributed by Al-Hayat Media Center, Daesh's media center. Dabiq, which now has fifteen issues, is written in English and was first published on July 5, 2014, only days after Abu Bakr Al-Baghdadi announced the reinstatement of the Caliphate in the Mosul mosque. In contrast, the French-language magazine Dar al-Islam, has ten issues, the first of which was released online on December 23, 2014. The fact that jihadist discourse appears so persuasive to many young people has raised questions about the effectiveness of institutional counter-discourse, which seems to have more difficulty in achieving its intended goals. In order to understand the factors determining the effectiveness of a discourse, it was necessary to build a corpus of counter-discourses, disseminated on the web by the French government and the European Union. The specificity of this multiple corpus, composed of four sub-corpora (French-speaking jihadist discourse, Englishspeaking jihadist discourse, French-speaking counter-discourse, and English-speaking counter-discourse), allowed us to conduct a contrastive analysis both between the two languages and between the two discourses. Adopting a quantitative-qualitative methodology, which relies on the use of textual analysis tools such as Tropes and Iramuteq, as well as on a more in-depth manual study, proved crucial. The combination of these two approaches allowed us to identify both the general characteristics and the specific features of each sub-corpus.

Results

Jihadist discourse, in both the French and English languages, is built around four concepts: threat, obligation, prohibition and reward. Since the magazines Dabiq and Dar al-Islam are aimed at an audience that has already adhered to the jihadist ideology, their objective is to incite the reader to act in the name of that ideology. The discourse of Daesh is thus built on a series of obligations and prohibitions that are presented to the reader as religious dictates. The reference to an unquestionable deity aims to legitimize the jihadist discourse and to lead readers to respect all the obligations and prohibitions that are imposed on them. In order to modulate readers' behavior and encourage them to respect these, the writer promises them rewards that they can only obtain by respecting all these indications. Specifically, these are rewards for adhering to the jihadist ideology, for adopting good behavior, and for any violent action taken against Daesh's enemies. These rewards include Allah's forgiveness, as well as glory and paradise. On the contrary, if the reader does not follow these dictates, he or she will face negative consequences that are presented and perceived as threats.

The qualitative analysis conducted on the way the discourse of Dabiq and Dar al-Islam revolves around these four notions (threat, obligation, prohibition, and reward) showed that English-speaking and French-speaking jihadist discourse follow the same rhetorical pattern. However, significant differences were identified between these two discourses. The Anglophone jihadist discourse focuses on the behavior that every Muslim should adopt. This is confirmed by the fact that the majority of the threats expressed in Dabiq are directed at any sympathizer who does not comply with the obligations and prohibitions imposed upon them. On the contrary, the Francophone jihadist discourse seems to focus not on the jihadist community but on the figure of the enemy. In Dar al-Islam, most of the threats refer to violent actions against the enemies of Daesh. In other words, the target of the message has a significant impact on the content of the message itself.

While the jihadist discourse follows a clear rhetorical pattern, the counter-discourse has no discursive structure. Differences can also be identified in the content of the four concepts. With regard to how the threat is expressed, unlike the jihadist discourse which presents the writer's community as the agent of the threat, both against its enemy and against its sympathizers, in the counter-discourse, the writer's community is shown as the victim of the jihadist threat. Furthermore, in the jihadist discourse, obligations and prohibitions indicate the behavior to be adopted in order to obtain Allah's forgiveness and access to paradise. In contrast, in the institutional counter-discourse, they indicate what to do in case of an attack. A further difference concerns rewards: unlike in the jihadist discourse, where rewards are intended to motivate the reader to act, in the counter-discourse this category is absent. The results from this qualitative analysis revealed that the jihadist discourse aims to incite the reader to take part in the conflict. In contrast, the institutional counter-discourse is limited to informing the reader, as it is only up to institutions to take action to address jihadist radicalization and terrorism. This difference between the two discourses could be the main factor that makes the jihadist discourse fascinating to its supporters, unlike the counter-discourse, which seems to have more difficulty in achieving its goals.

Faced with Daesh's propaganda, which modulates its discourse by proposing motives that adapt to and echo the different readers it addresses, the institutional discourse condemns jihadist action without proposing an alternative. If Daesh suggests to young people whose vocation is humanitarian work that they go to the Middle East to help Syrian children, the institutional discourse tends to focus on the plight of these children. However, this discourse is aimed at an audience that is looking for a cause to act on. Therefore, a message that presents the possibility of helping Syrian children who have fled the war in Syria and are now in Europe may prove more effective. In other words, through a targeted discourse, addressed to the different profiles likely to adhere to the ideology promoted by Daesh, the government could compete with the jihadist discourse. Moreover, the fact that the institutional counter-discourse tends to present the community as a victim of jihadist action risks feeding a feeling of hatred against Daesh and, in some cases, against migrants and Muslims. On the contrary, a discourse highlighting the values of the community and calling for unity would fuel a sense of "living together" and, consequently, to move away from any conflict and ideological polarization. In this perspective, the obituary portraits of the 159 victims of the Paris (November 2015) and Nice (July 2016) attacks published by Le Monde, could be considered as a potential counter-discourse4 .

Analysis perspectives

The case of Twitter Dabiq and Dar al-Islam are thus an important source of information on jihadist ideology and discourse. However, it is through social networks that Daesh can reach a much wider readership, facilitated by the fact that its sympathizers are no longer merely passive recipients of jihadist propaganda but can actively contribute to the production of extremist content5 . To deal with the proliferation of jihadist messages on social networks, several platforms, such as Facebook, Twitter and YouTube, have engaged in the detection and censorship of speeches that glorify terrorism and exalt the ideology promoted by Daesh. The techniques that are used to censor jihadist speech are based primarily on keyword detection.

However, the use of a pre-existing dictionary can have significant limitations. The fact that the same statement, and in some cases the same term, can be interpreted in several different ways makes it even more difficult to detect such speech. The term "jihad", for example, is perceived as something positive by Daesh sympathizers, as it would allow them to enter paradise, while it is perceived negatively by those who do not adhere to the jihadist ideology, as they tend to associate jihad with the attacks fomented by Daesh. Moreover, the same term is used by jihadists as well as the media, professionals and academics. It would therefore be necessary to adopt detection techniques that go beyond the simple lexicon. In this regard, a quantitative-qualitative analysis was conducted on a corpus of 4,000 tweets extracted from a series of hashtags and accounts that, in the literature on Daesh's use of social networks, had been associated with jihadist discourse. In order to analyze this corpus of tweets and to be able to compare it to the corpus of the two Daesh magazines, we adopted the same methodology that we applied to study the discourse of Dabiq and Dar al-Islam. Specifically, we wanted to identify the expression of threat, obligation, prohibition and reward in the tweets constituting our corpus. The objective here was to verify whether the discourse disseminated on Twitter follows the same rhetorical pattern on which the discourse of Dabiq and Dar al-Islam is built. This study revealed that the corpus of tweets presents the same components as the discourse of the two official Daesh magazines: threat, obligation, prohibition and reward. However, the production parameters imposed by Twitter, as well as the limited size of the corpus studied, do not allow for an advanced analysis of the discursive structure of the jihadist messages broadcast on this platform.

Limitations and avenues of analysis

While the study conducted on Dabiq and Dar al-Islam revealed the limitations of some linguistic analysis tools, the study conducted on the corpus of tweets showed the difficulties that this type of data can pose. As far as Dabiq and Dar al-Islam are concerned, our study has shown the limits of the analysis carried out with software that is based on a pre-established dictionary and therefore cannot detect the specific features of the text studied. Moreover, as they cannot take into account the point of view of the writer, these textual analysis tools do not allow the interpretation of the nuances and aims of the discourse. Developed from the limitations of these software programs, this qualitative analysis of jihadist discourse does, however, provide a basis for enriching the pre-existing dictionary or for establishing a lexicon specific to jihadist discourse. This would make it possible to refine the detection and categorization of discourse likely to have been produced by Daesh sympathizers. Different difficulties were encountered in the analysis of the corpus of tweets. When extracting tweets, the text is taken out of its context of production. Therefore, as these are decontextualized statements of no more than 140 characters, the text can evoke several interpretations. Unlike the jihadist discourse of the two official Daesh magazines, the tweets we examined cannot be linked to a jihadist context with certainty. Moreover, the specific features of this corpus make it difficult, if not impossible, to determine the point of view of the writer as well as to model a rhetorical pattern of jihadist discourse produced on Twitter.

The difficulties presented by this type of corpus suggest the need for a specific quantitative-qualitative study that could reveal specific features of jihadist discourse on social networks. From a quantitative point of view, a larger corpus would allow for machine learning and, consequently, for testing the results obtained. From a qualitative point of view, the contextualization of the tweets studied (user profile and chronology of photos and messages posted) would facilitate the distinction between jihadist discourse and any other discourse. The combination of these two approaches could thus reveal a rhetorical pattern of the discourse disseminated by Daesh on social networks that would allow the detection of jihadist content based not only on the lexicon used but also on their discursive structure. Furthermore, the comparison of the rhetorical pattern on which the jihadist discourse is built with other extremist discourses could lead to the modeling of a discursive structure specific to all extremist discourses.

Conclusion

This analysis of jihadist discourse and institutional counter-discourse has allowed us to identify elements that, until now, had gone unnoticed by other disciplines. We refer to the differences between Daesh discourse, which aims to incite the recipient to act, and counter-discourse, which, on the contrary, aims only to inform. Equally important are the differences between the French-speaking jihadist discourse, which focuses on the enemy, and the English-speaking jihadist discourse, which focuses on the jihadist community. These are factors that would need to be taken into account when different countries undertake collaboration to combat jihadist radicalization and terrorism. Considering the many findings from this research, we can say that linguistic analysis is crucial in the study of jihadist radicalization as well as in the development of a strategy to counter this phenomenon.

Translated by SLT Marc BOGARD and the French Gendarmerie Officers Academy Language Department

Bibliography

  BENSLAMA, Fethi. Un furieux désir de sacrifice : le surmusulman. Paris : Edition du Seuil, 2016. Homeland Security Institute. The Internet as a Terrorist Tool for Recruitment and Radicalization of Youth. Prepared for U.S. Department of Homeland Security. Science and Technology Directorate, 2009. KHOSROKHAVAR, Farhad. Radicalisation. Paris : Éditions de la maison des sciences de l'homme, 2014. LAMBERTI, Carolina. « Gli strumenti di contrasto al terrorismo e al cyber-terrorismo nel contesto europeo ». Rivista di Criminologia, Vittimologia e Sicurezza, 8(2), 2014, p. 138-161. RENAUT, Laurène, ASCONE, Laura. « Contre-discours au discours de haine jihadiste : de l'expression de la conflictualité à la fabrique du doute ». Semen, 47. Accepté pour publication, 2019. ASCONE, Laura. La radicalisation à travers l'expression des émotions sur Internet. PhD thesis in language sciences, University of Cergy-Pontoise, under the supervision of Julien LONGHI and Agnès CELLE, defended on 22 November 2018.

RENAUT, Laurène, ASCONE, Laura. « Contre-discours au discours de haine jihadiste : de l'expression de la conflictualité à la fabrique du doute ».Semen, 47, 2019.

Homeland Security Institute. The Internet as a Terrorist Tool for Recruitment and Radicalization of Youth. Prepared for U.S. Department of Homeland Security. Science and Technology Directorate, 2009.

Author : Laura Ascone Author : Laura Ascone Author : Laura Ascone

The content of this publication is to be considered as the author's own work and does not engage the responsibility of the CREOGN.


Daniel Mr

Guinier

M Philippe

SOCIAL NETWORKS: PARADOXES AND TECHNOLOGICAL CONVERGENCE

Doctor of Science, expert in cybercrime and financial crimes at the International Criminal Court in The Hague

In 2018, there were some 4.2 billion Internet users (i.e. 55% of the world's population), 3.4 billion of whom were users of social networks (Facebook, Twitter, LinkedIn, Whatsapp, etc.). In addition to their growing importance, social networks have multiple facets: a space for sharing and exchanging information, they are also proving to be places of influence and even manipulation. New forms of threats and vulnerabilities have emerged, particularly in relation to the manipulation of public opinion. The use of bots 2 and trolls 3 has also made it possible to industrialise the sharing of false information.

The multiple challenges of social networks

A medium of exchange subject to numerous abuses Social networks have gradually become a place for exchanging knowledge and information. On the borderline between private and public life, the law is enforced in respect of everyone's rights, despite the feeling of impunity that can sometimes prevail.

However, caution is needed on social networks, which are regularly used for manipulation or discrediting purposes. Any information disseminated can thus be instantly seen, used and retransmitted, including by people who were not necessarily the original recipients. This instantaneousness explains the growing phenomenon of the spread of rumours or false information (fake news). It is therefore more necessary than ever to measure the risk inherent in social networks in order to use them better.

A space of construction and vulnerabilities for users

Social networks have helped democratize access to public expression for populations that previously had no access to it. Today, people use them to reveal a lot of information about themselves, but also about other people. Each Internet user builds his own image on social networks, in the form of a digital portrait that he tends to enhance as he wishes.

This propensity of users to reveal themselves on social networks is sometimes out of step with the proximity they have with other users and is not always controlled. Many users do not care about the confidentiality of the A "bot" is a computer program that operates autonomously and whose main mission is to perform tasks repeatedly. Automated subscriptions, conversational and interactive robots (chatbot), etc., the bot relies on databases to respond to surfers' requests.

3 "Trolls" are accounts of surfers who voluntarily start controversies on social networking sites, discussion forums or blogs, with the aim of provoking or generating reactions in return, through excessive, insulting or advertising contributions.

information they publish and the access settings to their accounts and publications. Also, information is regularly treated too lightly and is perceived and retransmitted without any real control of its validity. In addition, a lot of information is passed from one social network to another, for example from Facebook to Twitter, which amplifies its reach.

Digital challenges for companies

Social networks are a recent sphere of communication and influence for companies. The forms of exchange that they induce have gradually led to the emergence of the issue of digital influence. The need to actively engage in social media is nowadays the guarantee for companies to play on equal terms with their competitors. Conversely, disregarding the digital sphere is in some ways equivalent to ceding a significant amount of notoriety and reputation to adversaries. This is all the more important because the immediacy of the dissemination of information makes it more necessary than ever to monitor the situation in order to ensure that the company can react quickly and appropriately in an increasing context of damage to its reputation. In fact, the latter is now perceived as a major risk4 , given the wide audience possibilities provided by the Internet. A good -or bad -reputation can thus spread to hundreds of millions of people in a very short time.

Companies are therefore trying to control their digital identity through their representation on social networks, without however being able to guarantee total control of their e-reputation, which can be undermined by certain major influencers. In fact, social networks have seen the emergence of new forms of vulnerabilities and threats for the various actors who evolve on them.

New threats and vulnerabilities

The online influence ecosystem

The online influence ecosystem is composed of several overlapping spheres, shared between human and robotic Internet users. Within this ecosystem, some great influencers stand out: renowned experts, media personalities, activist groups, etc. These modern "gurus" are followed by a host of "adepts" who echo back their messages.

The moderation system of social networks, based on algorithms, offers new possibilities to these influencers, whether they are human or robotic.

A complex algorithmic system

The moderation of social networks is essentially the responsibility of algorithms whose action on the reputation and therefore the influence of content is decisive, as in the case of Twitter, which gives priority to the tweets deemed most relevant. This algorithmic selection has immediately visible consequences: increase in the number of views, likes, retweets, etc.

Several questions arise as to how these algorithms work: firstly, on what criteria should the selection be made between what is relevant and what is not? Similarly, on what basis should the deletion of content or the suspension of an account be decided? A first issue thus lies in the legal qualification of the content. A second difficulty concerns the capacity to react and process the enormous flow of data. In this respect, nearly 10 million tweets are published every minute throughout the world. Finally, another major challenge lies in the vulnerability of the algorithms to possible attacks, whether bypassing or malicious modifications. Moreover, this threat is accentuated by the possibility of errors due to the complexity of the algorithms5 .

New vulnerabilities : bots and trolls

The development of social networks thus has its dark side. Their growth has allowed new forms of manipulation, particularly through the worldwide propagation of false or misleading information, which has been amplified by the use of trolls and bots. Trolls, social network accounts created by humans, are indeed a major vector for the propagation of biased content. Their use today is essentially aimed at creating controversy around targeted topics. Similarly, social bots are programmes designed to deceive Internet users by simulating human behaviour through automated interactions on social networks. By posing as real people, they are able to infiltrate social networks and gain the trust of users, sometimes for malicious purposes. They are also supported by classic bots, which are linked to a machine and make it possible to amplify the virality of targeted information. Bots and trolls are major distribution relays for certain influencers, acting as real sounding boards for social network users. The impacted profiles can be counted in the millions.

The case of the 2017 French presidential campaign

These disinformation operations were illustrated on a large scale during the 2017 French presidential campaign. E. Ferrara thus brought to light a vast operation to manipulate public opinion6 , by analysing 17 million tweets relating to the election period, posted between 27 April and 7 May 2017. In total, some 18,324 social bots are said to have participated in this disinformation campaign targeting, in particular, candidate Emmanuel Macron, with a peak of tweets attributed to bots on 7 May 2017, the date of the elections. However, it has been shown that these disinformation cascades are first attributed to human tweets, which are then amplified by a resonance phenomenon. This new threat posed by bots and trolls is now taking on a particular dimension, boosted by the new opportunities offered by technological convergence.

The possibilities brought about by technological convergence

Further improvement of information tools

Technological convergence, initially limited to computer technologies, now refers more broadly to the integration of several devices, services and networks within a single system or device. It is a ubiquitous phenomenon and involves combining technologies from several fields such as multimedia or telecommunications.

In the field of computer science, progress made in artificial intelligence has allowed for the advanced development of bots, enabling them to simulate real Internet users more than ever before, making them undetectable. This manipulation is accentuated by mixing false information with accurate and verifiable content. These influence strategies take the form of real destabilisation operations, targeting the very foundations of democracy7 .

The limits of technological convergence

However, this progress must be put into perspective, as demonstrated by the Tay experiment conducted by Microsoft in March 2018. A chatbot with artificial intelligence, Tay took on the profile of a teenage girl to chat on social networks, basing its words on pre-written answers or public databases. The experiment quickly gained momentum, with Tay attracting 23,000 followers in less than a day, sending out some 100,000 tweets. However, limits to the chatbot's learning and exchange capabilities soon became apparent after the robot made inappropriate or racist comments, and Tay was soon the target of insults and sexist remarks. Microsoft therefore finally put an end to the experiment.

What is the response of the digital players?

In contrast to the primary vocation of social networks as a place for sharing knowledge, technological convergence, by combining the Internet, social networks, bots, artificial intelligence and big data, is likely to represent a growing asset for operations to destabilise or manipulate public opinion. The challenge for social networking platforms will be to maintain rigorous ethics to guarantee the proper use of social networks, under the control of states and institutions. The latter have been able to measure the extent of the risks of manipulation, leading them to strengthen consultation between the various stakeholders concerned: public authorities, operators, social network users, etc. A change in legislation also merits in-depth examination, in order to settle the question of the balance between freedom of expression and control of the authenticity and truthfulness of comments posted online on social networks. The aim is to provide a definitive response to the recurrent accusations that public freedoms on the Internet are being undermined, particularly in terms of freedom of expression. In this respect, particular attention should be paid to Law 2018-1202 of 22 December 2018 on the fight against the manipulation of information 8 , particularly during the European electoral process.

Conclusion

Social networks constitute a single reality, but perceived from several different angles, depending on the nature of the actors involved: Internet users, companies, social network operators, etc. In fact, the ecosystem of online influence reveals divergent interests between these different actors, made up of human beings but also of "algorithmic entities", with increasing capacities for action and to which technological convergence has conferred new possibilities. Social networks therefore have a paradoxical nature that must be taken into account in order to avoid certain abuses, one of the manifestations of which is the manipulation of public opinion. Fake news and provocative messages are indeed a serious threat for all web actors, which should be addressed and regulated, either through legal tools or through a global cooperation process.

  Translated by SLT Clément DE SAVIGNY and the French Gendarmerie Officers Academy Language Department 8 General WATIN-AUGOUARD, Marc. Law n° 2018-1202 of 22 December 2018 on the fight against information manipulation. Note du CREOGN [online], n° 36, january 2019. Available on : https://www.gendarmerie.interieur.gouv.fr/crgn/Publications/Notes-du-CREOGN/La-loi-n-2018-1202du-22-decembre-2018-relative-a-la-lutte-contre-la-manipulation-de-l-information

Aon report 2017 and Deloitte study.

GUINIER, David. Sur la place des algorithmes et les exigences face à la complexité. Revue Experts, n° 139, july 2018, p. 40-44. (*"On the place of algorithms and the requirements of complexity")

FERRARA, Emilio. Disinformation and social bot operations in the run up to the 2017 French presidential election , University of Southern California, Information Sciences Institute, Vol. 22, n° 8.

JEANGÈNE-VILMER, Jean-Baptiste, ESCORCIA, Alexandre, GUILLAUME, Marine, HERRERA, Janaina. Les manipulations de l'information : un défi pour nos démocraties. ("Information manipulation: a challenge for our democracies.") Report of the Centre d'analyse, de prévision et de stratégie (CAPS) from the Ministry of Europe and Foreign Affairs and the Institut de recherche stratégique de l'École militaire (IRSEM), august 2018, p. 210.

Author : Philippe AUBERT-COUTURIER

The content of this publication is to be considered as the author's own work and does not engage the responsibility of the CREOGN.


In its desire to organize the cybersecurity sector, the European Union intends to promote a European digital market and stimulate the competitiveness of its companies. It also aims to defend its own interests, inherent to the economic, societal and security values it intends to preserve. As a "regulatory state", it is part of a global perspective of multi-level and multi-dimensional collaboration with both the Member States and the private sector. The modes of action are diverse, ranging from imposing legal obligations to creating dialogue platforms and awarding grants to finance projects in the field of research and innovation (R&I). The contractual public-private partnership (cPPP) is an example of a proactive policy led by the EU, where the focus is on R&I to organize and densify the European cybersecurity sector.

Public-private cooperation in cybersecurity is crucial. The Bockel parliamentary report of 2012 judged the level of cooperation with the private sector to be insufficient, even though companies are an essential link in the cybersecurity chain. One of the measures recommended was to increase the use of public funding for research and development in the field of information systems security. This is precisely the path that the European Union is going to take, whose involvement in cybersecurity was to be strengthened, according to the recommendations of the same report. There has been a major change in perception in this respect in recent years. Initially, cybersecurity was first associated with the security of information and communication networks, seen as a means to foster the growth of the digital economy. However, cybersecurity goes beyond digital security itself. It is not only a question of ensuring Europe's prosperity, by guaranteeing the proper functioning of the economic market, but also ensuring the defense of European values, in particular the preservation of democracy and fundamental rights. In other words, cybersecurity has a political and societal dimension. The 2013 cybersecurity strategy recalls two key elements in this regard: first, security is a shared responsibility; second, the private sector owns and operates significant parts of cyberspace 1 . As such, its inclusion is fundamental in the definition of a Unionwide cybersecurity policy, the manifestation of a European "regulatory state" in this field.

Including the private sector in an embryonic European cybersecurity

The protection of information systems dates back to the late 1990s. At the time, the measures taken were intended to promote the emergence of an information society in Europe. The action plan known as eEurope called for the establishment of a European framework for the protection of information systems. A communication from the European Commission in 2001 led to the establishment of an important legislative base in the fight against computer crime, a phenomenon whose victims are not only individuals, but also businesses 2 . A proposal presented in April 2002 led to the adoption in 2005 of a framework decision called "cyber attacks" aimed at approximating national criminal rules against attacks on information systems 3 .

Issue 29 December 2017

PUBLIC-PRIVATE COOPERATION AT THE EU LEVEL

The emergence of a European "regulatory state" for cybersecurity

By Pierre Berthelet, CREOGN associate researcher

The threat posed by cybercrime requires greater intervention, not only from the Union, but also from the private sector. At the operational level, efforts are focused on the fight against illegal content on the Internet, in particular against child pornography. They are translated into financial support for projects in this field. This operational cooperation, within the framework of concrete projects, is coupled with a more institutional cooperation. Various companies, including software manufacturers, are in a position to combat cybercrime, in particular attacks on information systems that are so serious as to jeopardize the achievement of a digital society. A Permanent Stakeholders Group (PSG) was set up for this purpose within the European Network and Information Security Agency (ENISA) to maintain regular dialogue between them and the private sector4 . The same applies to Europol. Europol's Anti-Cybercrime Unit (EC3) is an Internet investigation center at the service of Member States. Validated in 2012 by the EU Council, EC3 collaborates with the private sector. To this end, Europol has signed memorandums of understanding with several major information and communications companies to carry out its missions. The European cybersecurity strategy, which was approved the same year the EC3 was inaugurated, in 2013, places the private sector as the essential link in a security that can only be collective. Wishing to promote cyber-resilience in the Union, it considers that only an effective collaboration between public authorities and businesses can achieve this objective. In this sense, the Union is setting itself up as a regulatory body 5 . The "regulatory State" is in the wake of the "propulsive State " 6 described by Charles-Albert Morand, responsible for correcting market fluctuations and ensuring that major balances are maintained 7 . Preferring flexible law, it does not, however, shy away from the use of more traditional, i.e., more restrictive law. This ambivalence of the State as a producer of "neo-modern" law, mixing flexible regulation and more authoritarian regulation, is found from the point of view of the Union, in its modes of intervention. The regulatory State is placed in the perspective of transnational governance. The Union, as a regulatory State, is in fact part of the global perspective of a multi-level and multidimensional collaboration, both with the Member States and with the private sector 8 . The modes of public action are diverse, ranging from granting subsidies for projects in the field of research and innovation to increasing institutionalized dialogue, as well as imposing obligations in a more traditional regulatory framework. It is a matter of the Union working closely with the private sector, recognizing it as an essential partner in order to achieve the political goals it has set itself (fighting against cyber threats as part of the preservation of the growth of the European digital market), but also subjecting it to a series of legal obligations in this area. The most important of these is the Network and Information Security Directive (NIS Directive) 9 which compels operators of essential services and digital service providers to notify national authorities, in the event of cyber incidents.

Progressive organization of the European cybersecurity sector

A communication published on July 5, 2016, the day before the SRI Directive was adopted, aims to create industrial capabilities in the field of cybersecurity at the EU level 10 . This text has two dimensions that are two sides of the same coin: the obverse side highlights the private sector as a target of cyber threats. As the manager of certain critical infrastructures, it is vulnerable to cyber incidents, some of which largescale. The reverse side highlights the private sector as a provider of cyber security. This communication therefore intends to promote competitiveness and innovation. The idea is not so much to promote free competition as to ensure, on the contrary, regulation of the European cybersecurity sector, echoing this idea of the "regulatory state". Indeed, the Union suffers from a lack of interoperable solutions in case of cyber incidents. It is therefore a question of making up for the failures of the single market by taking a series of measures to structure this sector as well as possible. Noting the failure of the Lisbon strategy, the new strategy of March 2010, called "Europe 2020", intends to ensure the development of a strong industrial base. Among the seven initiatives retained by this strategy are the improvement of the competitiveness of the European Union's industry on a global scale and increased efforts in research and innovation. The Horizon 2020 program is part of this approach, as it aims to ensure the financing of research and innovation in the Union for the period 2014-2020. With a budget of 79 billion euros for the period 2014-2020, it is structured around three priorities, including "industrial primacy" and "societal challenges". To understand the emergence of a European cyber industry, it is important to go back to the 2004 attacks on the Atocha train station in Madrid. An action plan was approved in 2006 and a directive adopted in 2008 to strengthen the protection of critical infrastructures. The action plan recalls that the protection of this type of infrastructure is the responsibility of the Member States, but a concerted approach is necessary for infrastructures of European scope. The 2008 Directive establishes a Europe-wide procedure for the identification and designation of European Critical Infrastructures (ECI). Infrastructure related to the Information and Communication Technologies (ICT) sector is addressed in a 2009 Communication that sets out an action plan for the protection of "Critical Information Infrastructures" (CII)11 . Demonstrating the desire to move forward quickly, a communication was published in 2011 to take stock of the 2009 action plan12 . At this stage, however, there is no European action truly dedicated to cybersecurity, in the sense that CCI protection is included in an overall framework, with cybersecurity as its theme. The 2013 strategy dedicated to it aims to fill this gap. It establishes five priorities, including achieving cyber resilience and developing industrial and technological resources for cybersecurity.

As for the 2017 cybersecurity strategy, it deepens the efforts initiated since 2013. It notes that despite the positive results achieved in cybersecurity, the Union remains vulnerable to cyber incidents. It intends to strengthen cyber-resilience and promote competitiveness and innovation in the European cybersecurity sector. It is no coincidence that this strategy is being presented on the same day as the revisited strategy for the Union's industrial policy 13 . Such a strategy demonstrates the Union's will to enter a new industrial era marked by major technological breakthroughs, notably robotics, the Internet of Things, and artificial intelligence. Proposing to "make European industry stronger", it aims to invest in the industry of the future. Innovation is the engine of growth, which is why this strategy intends to promote research. Aiming to "modernize industry to bring it into the digital age", according to the terms used in the revised strategy, it is intended to promote the development of the Union's industrial capacities. The Contractual Public-Private Partnership (cPPP) is part of this approach. The cPPP aims to transcend the divisions and fragmentation inherent in the proliferation of actors, by seeking to coordinate actors from the public and private sectors, and located at different levels, European, national and local. It is characterized by an approach based on co-management, by the use of flexible law and by the networking of public and private actors (European Commission and agencies, ministries, local authorities, public research organizations, SMEs, universities and clusters). The idea of cPPPs, which was born in Strasbourg on July 5, 2016, is not new in itself. Several Member States have already opted for this type of public action instrument. For its part, the European Union has initiated a PPP in the field of resilience. This is the EP3R, established by the 2009 Communication and dedicated to the resilience of critical infrastructure. Mentioned by the Strategy for a Digital Single Market in Europe of May 6, 2015, the cPPP follows a similar logic to the IP3R: reconciling the expectations of the European institutions, which establish general policy priorities, and technical solutions from the private sector to best fit the priorities. The total budget of the cPPP amounts to 450 million euros. The goal is to reach 1.8 billion by 2020, thanks to the investment of the companies involved. Specified by the aforementioned communication of July 5, 2016, the creation of the cPPP is part of three complementary perspectives, the completion of the European digital market, by promoting competition between companies, the creation of a European industrial base in the field of cybersecurity and, finally, the defense of the Union's own interests. As a result, the cPPP intends, thanks to the multiplier effect of the amounts invested in R&I, to enable the development of a European cybersecurity sector that masters certain technologies to enable the Union to protect interests that it considers essential.

The 2017 cybersecurity strategy recalls that the level of investment in the United States amounted to $19 billion (€16 billion) for the year 2017, 35% more than in 2016. The cPPP is therefore a first step. It reflects an awareness not only on the part of the Union of the need to get involved in a proactive policy, but also on the part of the Member States, of the imperative to go beyond national approaches based on purely national or bilateral public-private partnerships. The cPPP is part of an era in which these partnerships are

Currently defined by Regulation (EU)

526/2013, this group includes representatives from the cybersecurity industry as well as academia. 5 On this thesis of the regulatory state in Europe, see Majone, G., La Communauté européenne : un État régulateur, Paris, Montchrestien, 1996. 6 Morand, C.-A., Le droit néo-moderne des politiques publiques, Paris, LGDJ, 1999. 7 Chevallier, J., « L'État régulateur », Revue française d'administration publique, vol. 111, n° 3, 2004, p. 473-482. 8 Chowdhury, N., Ramses A. Wessel, R. A., « Conceptualising Multilevel Regulation in the EU: A Legal Translation of Multilevel Governance? », European Law Journal, vol. 18, n° 3, mai 2012, p. 337-338. 9 Directive (EU) 2016/1148. 10 COM(2016) 410.

Author : Pierre Berthelet

COM(2009) 149, p. 6.

COM(2011) 163.

COM(2017) 479.

Author : Pierre Berthelet being conducted on a large scale, at a time when national and European interests, whether economic or societal, are more intertwined than ever.

Pierre Berthelet holds a doctorate in law, specializing in EU law, and is a security researcher at Laval University (Quebec). A former ministerial advisor, he is a member of the French Association of Security and Defense Law (AFDSD), a member of the Board of Directors of the French National Institute for Advanced Studies in Security and Justice (INHESJ), a member of the editorial board of the Cahiers de la Sécurité et de la Justice, and a research associate at CREOGN. He is the author of several books (including "Chaos international et sécurité globale. La sécurité en débats") and founder of the securiteinterieure.fr website.

Translated by SLT Guilhem SERENE and the French Gendarmerie Officers Academy Language Department Author : Pierre Berthelet

The content of this publication is to be considered as the author's own work and does not engage the responsibility of the CREOGN.


Michel Duprez
email: michel.duprez@inria.fr

Vanessa Lleras
email: vanessa.lleras@umontpellier.fr

Alexei Lozinski
email: alexei.lozinski@univ-fcomte.fr

Killian Vuillemot
email: killian.vuillemot@umontpellier.fr

ϕ-FEM for the heat equation: optimal convergence on unfitted meshes in space

Thanks to a finite element method, we solve numerically parabolic partial differential equations on complex domains by avoiding the mesh generation, using a regular background mesh, not fitting the domain and its real boundary exactly. Our technique follows the ϕ-FEM paradigm, which supposes that the domain is given by a level-set function. In this paper, we prove a priori error estimates in l 2 (H 1 ) and l ∞ (L 2 ) norms for an implicit Euler discretization in time. We give numerical illustrations to highlight the performances of ϕ-FEM, which combines optimal convergence accuracy, easy implementation process and fastness.

Introduction

The classical finite element method for elliptic and parabolic problems (see e.g. [START_REF] Thomée | Galerkin finite element methods for parabolic problems[END_REF]) needs a computational mesh fitting the boundary of the physical domain. In some applications in engineering or bio-mechanics, the construction of such meshes may be very time-consuming or even impossible. Alternative approaches, such as Fictitious Domain [START_REF] Glowinski | A fictitious domain method for Dirichlet problem and applications[END_REF] or Immersed Boundary Methods (IBM) (see e.g. [START_REF] Mittal | Immersed boundary methods[END_REF] for a review), can work on unfitted meshes but are usually not very precise. More recent variants, such as CutFEM [START_REF] Annavarapu | A robust Nitsche's formulation for interface problems[END_REF], demonstrate optimal convergence orders but are less straightforward to implement than the original IBM. In particular, CutFEM needs special quadrature rules on the cells cut by the boundary. Finally, we can also mention the Shifted Boundary Method [START_REF] Main | The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems[END_REF] that avoids the non-trivial integration by introducing a boundary correction based on a Taylor expansion.

A new Finite Element Method on unfitted meshes, named ϕ-FEM, combining the optimal convergence and the ease of implementation, was recently proposed in [START_REF] Duprez | ϕ-FEM: a finite element method on domains defined by levelsets[END_REF][START_REF] Duprez | A new ϕ-FEM approach for problems with natural boundary conditions[END_REF]. Initially developed for stationary elliptic PDEs, it has been extended in [START_REF] Cotin | ϕ-FEM: an efficient simulation tool using simple meshes for problems in structure mechanics and heat transfer[END_REF] to a broader class of equations, including the time-dependent parabolic problems, without any theoretical analysis. The goal of the present note is to provide such an analysis in the case of the Heat-Dirichlet problem

∂ t u -∆u = f in Ω × (0, T ), u = 0 on Γ × (0, T ), u |t=0 = u 0 in Ω, (1)
where T > 0, Ω ⊂ R d , d = 2, 3 is a bounded domain with a smooth boundary Γ given by a level-set function on R d Ω := {ϕ < 0} and Γ := {ϕ = 0} .

(Note that some FEM on unfitted meshes have been developed for such problems for example, in [START_REF] Schwartz | A Cartesian grid embedded boundary method for the heat equation and Poisson's equation in three dimensions[END_REF][START_REF] Mccorquodale | A Cartesian grid embedded boundary method for the heat equation on irregular domains[END_REF]).

For the discretization in time, we use the implicit Euler scheme. The Dirichlet boundary conditions are imposed via a product with the level-set function ϕ. An appropriate stabilization is introduced to the finite element discretization to obtain well-posed problems. A somewhat unexpected feature of this stabilization is that it works under the constraint on the steps in time and space of the type ∆t ⩾ ch 2 . This does not affect the practical interest of the scheme since it is normally intended to be used in the regime ∆t ∼ h. We shall provide a priori error estimates for this scheme in l 2 (H 1 ) norms of similar orders as for the standard FEM, cf. [START_REF] Thomée | Galerkin finite element methods for parabolic problems[END_REF]. We also study the l ∞ (L 2 ) convergence and prove a slightly suboptimal theoretical bound for it, while it turns out to be optimal numerically.

2 Definitions, assumptions, description of the scheme and the main result.

We assume that Ω lies inside a box O ⊂ R d and that Ω and Γ are given by [START_REF] Glowinski | A fictitious domain method for Dirichlet problem and applications[END_REF]. The box O is covered by a simple quasi-uniform simplicial (typically Cartesian) background mesh denoted by T O h . We introduce the active computational mesh

T h := T ∈ T O h : T ∩ {ϕ h < 0} ̸ = ∅ on Ω h = (∪ T ∈T h T )
o , the subdomain of O composed of mesh cells intersecting Ω, cf. Fig. 1 (right). Here, ϕ h is a piecewise polynomial interpolation of ϕ in finite element space of degree l ∈ N * on T O h . We shall also need a submesh T Γ h , containing the elements of T h that are cut by the approximate boundary Γ h := {ϕ h = 0}:

T Γ h = {T ∈ T h : T ∩ Γ h ̸ = ∅}.
Finally, we denote by F Γ h the set of the internal facets E of mesh T h belonging to the cells of the set

T Γ h , F Γ h := {E (internal facet of T h ) such that ∃ T ∈ T h : T ∩ Γ h ̸ = ∅ and E ∈ ∂T }.
Introduce a uniform partition of [0, T ] into time steps 0 = t 0 < t 1 < . . . < t N = T with t n = n∆t. The basic idea of ϕ-FEM is to introduce the new unknown w = w(x, t) and to set u = ϕw so that the Dirichlet condition u = 0 is automatically satisfied on Γ since ϕ vanishes there. Using an implicit Euler scheme to discretize (1) in time and denoting f n (•) = f (•, t n ), we get the following discretization in time: given u n = ϕw n find u n+1 = ϕw n+1 such that

ϕw n+1 -ϕw n ∆t -∆(ϕw n+1 ) = f n+1 . (3)
To discretize in space, we introduce the finite element space of degree k on Ω h ,

V (k) h = {v h ∈ H 1 (Ω h ) : v h | T ∈ P k (T ), ∀ T ∈ T h } ,
for some k ⩾ 1. Supposing that f and u 0 are actually well defined on Ω h (rather than on Ω only), we can finally introduce the ϕ-FEM scheme for (1) as follows: find

w n+1 h ∈ V (k) h , n = 0, 1, . . . , N -1 such that for all v h ∈ V (k) h Ω h ϕ h w n+1 h ∆t ϕ h v h + Ω h ∇(ϕ h w n+1 h ) • ∇(ϕ h v h ) - ∂Ω h ∂ ∂n (ϕ h w n+1 h )ϕ h v h + σh E∈F Γ h E ∂(ϕ h w n+1 h ) ∂n ∂(ϕ h v h ) ∂n -σh 2 K∈T Γ h K ϕ h w n+1 h ∆t -∆(ϕ h w n+1 h ) ∆(ϕ h v h ) = Ω h u n h ∆t + f n+1 ϕ h v h -σh 2 K∈T Γ h K u n h ∆t + f n+1 ∆(ϕ h v h ) (4) with u n h = ϕ h w n h for n ⩾ 1 and u 0 h ∈ V (k) h
an interpolant of u 0 . Moreover, ϕ h is the piecewise polynomial interpolation of ϕ in V (l) h , with l ⩾ k. This scheme contains two stabilization terms: the ghost stabilization (the sum on the facets in F Γ h ) as in [START_REF] Burman | Ghost penalty[END_REF], and a least-square stabilization (the terms multiplied by σh 2 ) that reinforces (3) on the cells of T Γ h .

Remark 1. Our approach can be easily generalized to non-homogeneous Dirichlet boundary conditions u = u D on Γ × (0, T ). We can pose then u n h = ϕ h w n h + I h u g (•, t n ) where u g is some lifting of u D from Γ to Ω h and I h stands for a finite element interpolation to V (k) h . Scheme (4) should then be modified accordingly, replacing ϕ h w n+1 h by ϕ h w n+1 h + I h u g (•, t n+1 ) which results in some additional terms on the right-hand side.

We recall from [START_REF] Duprez | ϕ-FEM: a finite element method on domains defined by levelsets[END_REF] the assumptions on the domain and on the mesh required in the theoretical study of the convergence of the ϕ-FEM scheme. These assumptions are satisfied if the boundary Γ is regular enough and the mesh T h is fine enough. Assumption 2. The approximate boundary Γ h = {ϕ h = 0} can be covered by element patches {Π k } r=1,...,NΠ such that :

• Each patch Π r can be written Π r = Π Γ r ∪ T r with Π Γ r ⊂ T Γ h and T r ∈ T h \ T Γ h . Moreover Π r contains less than M elements and these elements are connected; andw n h be the solution to (4) for n = 1, . . . , N . For σ large enough, there exist c, C > 0 depending only on the regularity of mesh T h and on the constants of Ass. 1-2 (with C also depending on

• T Γ h = ∪ r=1,...,NΠ Π Γ r ; • Two patches Π r and Π s are disjoint if r ̸ = s. Theorem 1. Assume Ω ⊂ Ω h , l ≥ k, Assumption 1-2, f ∈ H 1 (0, T ; H k-1 (Ω h )) and u ∈ H 2 (0, T ; H k-1 (Ω)) being the exact solution to (1), u n (•) = u(•, t n )
T ), such that if ∆t ⩾ ch 2 then N n=0 ∆t|u n -ϕ h w n h | 2 H 1 (Ω) 1 2 ⩽ C∥u 0 -u 0 h ∥ L 2 (Ω h ) + C(h k + ∆t) ∥u∥ H 2 (0,T ;H k-1 (Ω)) + ∥f ∥ H 1 (0,T ;H k-1 (Ω h ))
and

max 1⩽n⩽N ∥u n -ϕ h w n h ∥ L 2 (Ω) ⩽ C∥u 0 -u 0 h ∥ L 2 (Ω h ) + C(h k+ 1 2 + ∆t) ∥u∥ H 2 (0,T ;H k-1 (Ω)) + ∥f ∥ H 1 (0,T ;H k-1 (Ω h )) .
Remark 2.

• If k = 1, the norms on the right hand side of the estimates above can be replaced by the norm of f alone in

H 1 (0, T ; L 2 (Ω h )). Indeed, recalling Ω ⊂ Ω h , this assumption on f implies u ∈ H 2 (0, T ; L 2 (Ω)) ∩ H 1 (0, T ; H 2 (Ω))
, see e.g. [12, Theorems 5 and 6, Chapter 7.1]. On the other hand, imposing such regularity on u over Ω, would not suffice to control the extension of f outside of Ω, so that the regularity of f on Ω h should be postulated any way. This contrasts with the usual a priori estimates for standard FEM (see e.g. [START_REF] Thomée | Galerkin finite element methods for parabolic problems[END_REF]).

• If k > 1, we need to suppose the regularity of both u and f as stated above.

In the rest of the paper, the letter C, eventually with subscripts, will stand for various constants depending on the mesh regularity, the constants from Ass. 1-2, and also on T (when specifically mentioned). Before the proof of Theorem 1, we recall some results from [START_REF] Duprez | ϕ-FEM: a finite element method on domains defined by levelsets[END_REF] about ϕ-FEM for the Poisson equation with Dirichlet boundary conditions.

Lemma 1 (cf. [START_REF] Duprez | ϕ-FEM: a finite element method on domains defined by levelsets[END_REF]Lemma 3.7]). Consider the bilinear form

a h (u, v) = Ω h ∇u • ∇v - ∂Ω h ∂u ∂n v + σh E∈F Γ h E ∂u ∂n ∂v ∂n + K∈T Γ h σh 2 K ∆u ∆v.
Provided σ is chosen big enough, there exists an h-independent constant α > 0 such that

a h (ϕ h v h , ϕ h v h ) ⩾ α|ϕ h v h | 2 H 1 (Ω h ) , ∀v h ∈ V (k) h . Lemma 2 (cf. [6, Theorem 2.3]). For any f ∈ H k-1 (Ω h ), let w h ∈ V (k) h
be the solution to

a h (ϕ h w h , ϕ h v h ) = Ω h f ϕ h v h -σh 2 K∈T Γ h K f ∆(ϕ h v h )
and u ∈ H k+1 (Ω) be the solution to

-∆u = f in Ω, u = 0 on Γ extended to ũ ∈ H k+1 (Ω h ) so that u = ũ on Ω and ∥ũ∥ H k+1 (Ω h ) ⩽ C∥u∥ H k+1 (Ω) ⩽ C∥f ∥ H k-1 (Ω h ) . Provided σ is chosen big enough, there exists an h-independent constant C > 0 such that |ũ -ϕ h w h | H 1 (Ω h ) ⩽ Ch k ∥f ∥ H k-1 (Ω h ) and ∥ũ -ϕ h w h ∥ L 2 (Ω h ) ⩽ Ch k+ 1 2 ∥f ∥ H k-1 (Ω h ) .
Remark 3. This result is proven in [START_REF] Duprez | ϕ-FEM: a finite element method on domains defined by levelsets[END_REF] under the more stringent assumption f ∈ H k (Ω h ) which was used to assure ũ ∈ H k+2 (Ω h ) and to provide an interpolation error of ũ by a product ϕ h w h . However, in [13, Lemma 6] we have proven a better interpolation estimate ∥ũ -

ϕ h I h w∥ H s (Ω h ) ⩽ Ch k+1-s ∥f ∥ H k-1 (Ω h ) (s = 0, 1) for ũ = ϕw and the Scott-Zhang interpolant I h . Thus, f ∈ H k-1 (Ω h ) is actually sufficient. Lemma 3. For all v h ∈ V (k)
h , there holds

∥ϕ h v h ∥ L 2 (Ω h ) ⩽ C P |ϕ h v h | H 1 (Ω h ) .
Proof. Let Ωh = {ϕ h < 0}. By the Poincaré inequality,

∥ϕ h v h ∥ L 2 ( Ωh ) ⩽ Cdiam( Ωh )|ϕ h v h | H 1 ( Ωh ) ,
and diam( Ωh ) ⩽ diam(O). Moreover, thanks to [6, Lemma 3.4], it holds

∥ϕ h v h ∥ L 2 (Ω h \ Ωh ) ⩽ ∥ϕ h v h ∥ L 2 (Ω Γ h ) ⩽ Ch|ϕ h v h | H 1 (Ω Γ h ) ,
where Ω Γ h is the domain occupied by the mesh

T Γ h . We conclude noting Ω ⊂ Ωh ∪ Ω Γ h . Proof of Theorem 1. There exists a function ũ ∈ H 2 (0, T ; H k-1 (Ω h )), an extension of u to Ω h , such that ∥ũ∥ H 2 (0,T ;H k-1 (Ω h )) ⩽ C∥u∥ H 2 (0,T ;H k-1 (Ω)) . (5)
Let w n h be the solution to our scheme, which we rewrite as

Ω h ϕ h w n+1 h -w n h ∆t ϕ h v h + a h (ϕ h w n+1 h , ϕ h v h ) - T ∈T Γ h σh 2 T ϕ h w n+1 h -w n h ∆t ∆(ϕ h v h ) = Ω h f n+1 ϕ h v h - T ∈T Γ h σh 2 T f n+1 ∆(ϕ h v h ) (6)
for n ⩾ 1 while ϕ h w 0 h should be replaced with u 0 h for n = 0. For any time t ∈ [0, T ], introduce wh (

•, t) = wh ∈ V (k) h , as in Lemma 2, with f replaced by f -∂ t ũ evaluated at time t: a h (ϕ h wh , ϕ h v h ) = Ω h (f -∂ t ũ)ϕ h v h -σh 2 K∈T Γ h K (f -∂ t ũ)∆(ϕ h v h ). ( 7
)
Let wn h = wh (t n ) and e n h := ϕ h (w n h -wn h ) for n ⩾ 1 and e 0 h := u 0 h -ϕ h w0 h . Taking the difference between ( 6) and ( 7) at time t n+1 , we get

Ω h e n+1 h -e n h ∆t ϕ h v h + a h (e n+1 h , ϕ h v h ) - T ∈T Γ h σh 2 T e n+1 h -e n h ∆t ∆(ϕ h v h ) = Ω h ∂ t ũn+1 -ϕ h wn+1 h -wn h ∆t ϕ h v h - T ∈T Γ h σh 2 T ∂ t ũn+1 -ϕ h wn+1 h -wn h ∆t ∆(ϕ h v h ).
Taking

v h = w n+1 h -wn+1 h , i.e. ϕ h v h = e n+1 h
, applying the equality

∥e n+1 h ∥ 2 L 2 (Ω h ) -(e n h , e n+1 h ) L 2 (Ω h ) = ∥e n+1 h ∥ 2 L 2 (Ω h ) -∥e n h ∥ 2 L 2 (Ω h ) + ∥e n+1 h -e n h ∥ 2 L 2 (Ω h ) 2 ,
and estimating the terms in the RHS by Cauchy-Schwarz and inverse inequalities

∥∆e n+1 h ∥ L 2 (T ) ⩽ Ch -2 ∥e n+1 h ∥ L 2 (T )
we deduce that

∥e n+1 h ∥ 2 L 2 (Ω h ) -∥e n h ∥ 2 L 2 (Ω h ) + ∥e n+1 h -e n h ∥ 2 L 2 (Ω h ) 2∆t + (I) a h (e n+1 h , e n+1 h ) - (II) σh 2 Ω Γ h e n+1 h -e n h ∆t ∆e n+1 h ⩽ C ∂ t ũn+1 -ϕ h wn+1 h -wn h ∆t L 2 (Ω h ) ∥e n+1 h ∥ L 2 (Ω h ) (III)
.

Thanks to the coercivity lemma 1, the term (I) can be bounded from below by α|e n+1 h | 2 H 1 (Ω h ) . We now use the Young inequality (with some ε > 0) and the inverse inequality ∥∆e n+1 h

∥ L 2 (T ) ⩽ C I h -1 |e n+1 h | H 1 (T ) to bound the term (II): (I) -(II) ⩾ α|e n+1 h | 2 H 1 (Ω h ) - σh 2 2ϵ(∆t) 2 ∥e n+1 h -e n h ∥ 2 L 2 (Ω Γ h ) - ϵσC 2 I 2 |e n+1 h | 2 H 1 (Ω Γ h ) ⩾ 3 4 α|e n+1 h | 2 H 1 (Ω h ) - 1 2∆t ∥e n+1 h -e n h ∥ 2 L 2 (Ω Γ h ) , (9)
where we have chosen ϵ so that ϵσC 2 I /2 = α/4 and then assumed σh 2 /(ϵ∆t) ⩽ 1. This will allow us to control the negative term above by the similar positive term in [START_REF] Cotin | ϕ-FEM: an efficient simulation tool using simple meshes for problems in structure mechanics and heat transfer[END_REF], and leads to the restriction ∆t ⩾ ch 2 with c = σ/ϵ.

We turn now to the RHS of (8), i.e. term (III). By triangle inequality

∂ t ũn+1 -ϕ h wn+1 h -wn h ∆t L 2 (Ω h ) ⩽ ∂ t ũn+1 - ũn+1 -ũn ∆t L 2 (Ω h ) + ũn+1 -ũn ∆t -ϕ h wn+1 h -wn h ∆t L 2 (Ω h ) . (10)
By Taylor's theorem with integral remainder

ũn (•) = ũn+1 (•) -∆t∂ t ũn+1 (•) - tn+1 tn ∂ tt ũ(t, •)(t n -t)dt so that ∂ t ũn+1 - ũn+1 -ũn ∆t L 2 (Ω h ) = 1 ∆t tn+1 tn ∂ tt ũ(t, •)(t n -t)dt L 2 (Ω h ) ⩽ √ ∆t∥∂ tt ũ∥ L 2 (tn,tn+1;L 2 (Ω h )) .
Differentiating -∆u = f -∂ t u and ( 7) in time, we obtain thanks to Lemma 2,

∥∂ t (ũ(t) -ϕ h wh )(t)∥ L 2 (Ω h ) ⩽ Ch k+ 1 2 ∥(∂ t f -∂ tt ũ)(t)∥ H k-1 (Ω h ) .
Thus, for the second term in [START_REF] Mccorquodale | A Cartesian grid embedded boundary method for the heat equation on irregular domains[END_REF], we get by the last interpolation estimate:

ũn+1 -ũn ∆t -ϕ h wn+1 h -wn h ∆t L 2 (Ω h ) = 1 ∆t tn+1 tn ∂ t (ũ(t, •) -ϕ h wh (t, •))dt L 2 (Ω h ) ⩽ Ch k+ 1 2 √ ∆t ∥∂ t f -∂ tt ũ∥ L 2 (tn,tn+1;H k-1 (Ω h )) .
Collecting these estimates and applying the Young inequality with some δ > 0 and Poincaré inequality from Lemma 3, we get

(III) ⩽ C δ ∆t∥∂ tt ũ∥ 2 L 2 (tn,tn+1;L 2 (Ω h )) + h 2k+1 ∆t ∥∂ t f -∂ tt ũ∥ 2 L 2 (tn,tn+1;H k-1 (Ω h )) + δC 2 P 2 |e n+1 h | 2 H 1 (Ω h ) . (11)
Substituting ( 9) and ( 11) to ( 8) and taking δ so that δC 2 P = α/2 yields

∥e n+1 h ∥ 2 L 2 (Ω h ) -∥e n h ∥ 2 L 2 (Ω h ) 2∆t + α 2 |e n+1 h | 2 H 1 (Ω h ) ⩽ C ∆t∥∂ tt ũ∥ 2 L 2 (tn,tn+1;L 2 (Ω h )) + h 2k+1 ∆t ∥∂ t f -∂ tt ũ∥ 2 L 2 (tn,tn+1;H k-1 (Ω h )) .
Multiplying this by 2∆t and summing on n = 0, . . . , N -1, we get

∥e N h ∥ 2 L 2 (Ω h ) + α∆t N n=1 |e n h | 2 H 1 (Ω h ) ⩽ ∥e 0 h ∥ 2 L 2 (Ω h ) + C(∆t 2 ∥∂ tt ũ∥ 2 L 2 (0,T ;L 2 (Ω h )) + h 2k+1 ∥∂ t f -∂ tt ũ∥ 2 L 2 (0,T ;H k-1 (Ω h ))
). Thus, observing that the sum above can be stopped at any number n ⩽ N , we get max n=1,...,N

∥e n h ∥ L 2 (Ω h ) + ∆t N n=1 |e n h | 2 H 1 (Ω h ) 1 2 ⩽ C∥e 0 h ∥ L 2 (Ω h ) + C ∆t∥∂ tt ũ∥ L 2 (0,T ;L 2 (Ω h )) + h k+ 1 2 ∥∂ t f -∂ tt ũ∥ L 2 (0,T ;H k-1 (Ω h )) .
Lemma 2 applied to -∆u = f -∂ t u in Ω at times t n gives max n=0,...,N

∥ũ n -ϕ h wn h ∥ L 2 (Ω h ) ⩽ Ch k+1/2 ∥f -∂ t ũ∥ C([0,T ],H k-1 (Ω h )) , ∆t N n=1 |ũ n -ϕ h wn h | 2 H 1 (Ω h ) 1 2 ⩽ Ch k ∥f -∂ t ũ∥ C([0,T ],H k-1 (Ω h )) .
In particular,

∥e 0 h ∥ L 2 (Ω h ) ⩽ ∥u 0 -u 0 h ∥ L 2 (Ω h ) + ∥u 0 -ϕ h w0 h ∥ L 2 (Ω h ) ⩽ ∥u 0 -u 0 h ∥ L 2 (Ω h ) + Ch k+1/2 ∥f -∂ t ũ∥ C([0,T ],H k-1 (Ω h ))
. Combining this with the regularity of f and ũ, cf. ( 5), together with the bound ∥ • ∥ C([0,T ],•) ⩽ C∥ • ∥ H1 (0,T ;•) (with C depending on T ) gives the announced result .

Numerical experiments

In this section, we illustrate the performance of our approach on two test cases 1 . We have implemented ϕ-FEM in FEniCS [START_REF] Alnaes | Archive of numerical software: The fenics project version 1[END_REF], the codes of the simulations are available in the github repository In our numerical simulations, if the expected convergence is of order C 1 h p + C 2 ∆t m , we will fix ∆t = h p/m in such a way we only need to observe if the error is of order h p numerically.

Remark 4 (Norms for the simulations). To illustrate the convergence of the methods with the simulations, since it is numerically complex to compute the error on the exact domain Ω, we will use the following formula

∥u h -u ref ∥ 2 l 2 (0,T,H 1 0 (Ω ref )) ∥u ref ∥ 2 l 2 (0,T,H 1 0 (Ω ref )) ≈ N n=0 ∆t Ω ref |∇u h (., t n ) -∇u ref (., t n )| 2 dx N n=0 ∆t Ω ref |∇u ref (., t n )| 2 dx
,

and ∥u h -u ref ∥ 2 l ∞ (0,T,L 2 (Ω ref )) ∥u ref ∥ 2 l ∞ (0,T,L 2 (Ω ref )) ≈ max n=0,...,N Ω ref (u h (., t n ) -u ref (., t n )) 2 dx max n=0,...,N Ω ref (u ref (., t n )) 2 dx ,
where u h denotes an approximation of the L 2 -orthogonal projection of the solution on the reference mesh Ω ref and u ref the reference solution.

First test case : the source term is deduced from a manufactured solution and the FEM solution is compared to this manufactured solution. For this case, we will consider a simple smooth domain : the circle centered in (0, 0), with radius 1 as represented in Fig. 1. The level-set function is given using the equation of the circle, i.e. ϕ(x, y) = -1 + x 2 + y 2 . Its approximation ϕ h will be the interpolation of ϕ with P k+1 finite elements, except for Fig. 6 (right). Moreover, we consider the manufactured solution given by

u ref = cos 1 2 π(x 2 + y 2 ) exp(x) sin(t) so that u ref satisfies u ref (t = 0) = u 0 ref = 0 and u ref = 0 on Γ × (0, T ).
Here, Ω ref = Ω h . We represent the errors in l 2 (H 1 ) norm on Fig. 2 and in l ∞ (L 2 ) norm on Fig. 3, both with P 1 and P 2 finite elements (k = 1 and k = 2). Here, the numerical results fit well the theoretical convergence order of Theorem 1 and behaves even better since we observe a convergence of orders two and three for the l ∞ (L 2 ) norm instead of 1.5 and 2.5 respectively. We remark that the theoretical constraint ∆t ⩾ ch 2 is not satisfied for the P 2 finite elements but it does not affect the practical convergence. We also represent the l 2 (H 1 ) and l ∞ (L 2 ) errors with respect to the computation time (here, the computation time is the sum of time needed to assemble the finite element matrix 10 -1.5 10 -1 10 -0.5

10 -2 10 -1 1 1 h ∥u ref -u h ∥ l 2 ( H 1
)

∥u ref ∥ l 2 (H 1 ) ϕ-FEM Standard FEM 10 -1.5 10 -1 10 -0.5 10 -4 10 -3 10 -2 10 -1 1 2 h ∥u ref -u h ∥ l 2 ( H 1
)

∥u ref ∥ l 2 (H 1 ) ϕ-FEM Standard FEM
Figure 2: First test case. l 2 (0, T ; H 1 (Ω)) relative errors with respect to h with P 1 elements and ∆t = h (left) and with P 2 elements and ∆t = h 2 (right). Standard FEM (red squares) and ϕ-FEM (blue dots), σ = 1.

10 -1.5 10 -1 10 -0.5

h maxt i ∥u ref (ti)-u h (ti)∥0,Ω maxt i ∥u ref (ti)∥0,Ω ϕ-FEM Standard FEM 10 -1 10 -0.5 10 -5 10 -3 10 -1 1 3 1 2 h maxt i ∥u ref (ti)-u h (ti)∥0,Ω maxt i ∥u ref (ti)∥0,Ω ϕ-FEM Standard FEM 10 -4 10 -3 10 -2 10 -1 1 2 1 1.5
Figure 3: First test case. l ∞ (0, T ; L 2 (Ω)) relative errors with respect to h with P 1 elements and ∆t = h 2 (left) and with P 2 elements and ∆t = h 3 (right). Standard FEM (red squares) and ϕ-FEM (blue dots), σ = 1.

and to solve the finite element systems at each time step, without the time used to construct the meshes) in Fig. 4. We observe that in this case, ϕ-FEM is significantly faster than a standard FEM to obtain a solution with the same precision. In Fig. 5 (left), we represent the l 2 (H 1 ) error and in Fig. 5 (right) the l ∞ (L 2 ) error, both with respect to σ. This allows us to emphasize the influence of σ on the stability of the errors and validates our choice of σ = 1 in the other simulations.

Finally, in Fig. 6, we justify our choice for the degree of interpolation of ϕ since in our theoretical result, P k is sufficient but we observe here that the error decreases for l = 2. Furthermore, in our previous paper [START_REF] Duprez | A new ϕ-FEM approach for problems with natural boundary conditions[END_REF], our theoretical results in the Neumann case hold true only for l ≥ k +1. Here, since the interpolation is exact from l = 2 we do not need to compute highest degrees of interpolation for the level-set function to compare the results.

Second test case : the source term is given and the FEM solution is compared to a standard FEM solution on a very fine mesh. We now consider a more realistic test case Computation time (s) since we will apply some forces and consider the resulting distribution of heat in the considered domain. More precisely, this time, we impose u = 0 on Γ × (0, T ), the initial condition is u 0 = 0 in Ω and we define a source term given by f (x, y, z, t) = exp -

maxt i ∥u ref (ti)-u h (ti)∥0,Ω ref maxt i ∥u ref (ti)∥0,Ω ref ϕ-FEM Standard FEM
h maxt i ∥u ref (ti)-u h (ti)∥0,Ω ref maxt i ∥u ref (ti)∥0,Ω ref l = 1 l = 2
(x-µ1) 2 +(y-µ2) 2 +(z-µ3) 2 2σ 2 0 for each (x, y, z, t) ∈ Ω × (0, T ), with (µ 1 , µ 2 , µ 3 , σ 0 ) = (0.2, 0.3, -0.1, 0.3).
The final time is fixed to T = 1. Moreover, for this test case, we will consider a more complex and 3D domain from [START_REF] Burman | Cutfem: discretizing geometry and partial differential equations[END_REF], given by

ϕ(x, y, z) = x 2 + y 2 + z 2 -r 2 0 -A 11 k=0 exp - (x -x k ) 2 + (y -y k ) 2 + (z -z k ) 2 σ 2 0 , with (x k , y k , z k ) = r 0 √ 5 2 cos 2kπ 5 , 2 sin 2kπ 5 , 1 , 0 ⩽ k ⩽ 4 , (x k , y k , z k ) = r 0 √ 5 2 cos (2(k -5) -1)π 5 , 2 sin (2(k -5) -1)π 5 , -1 , 5 ⩽ k ⩽ 9 , (x k , y k , z k ) = (0, 0, r 0 ) , k = 10 , (x k , y k , z k ) = (0, 0, -r 0 ) , k = 11 ,
with r 0 = 0.6, σ = 0.3 and A = 1.5. The resulting domain and meshes are given in Fig. 7.

Here, u ref denotes the solution of a classical finite element method on Ω ref that is a very fine conforming mesh. In this case, to be more precise, we introduce a partition of the interval [0, T ] into time steps 0

= t ref 0 < t ref 1 < • • • < t ref M = T with t ref n = n∆t ref and ∆t ref = h p/m
ref , where h ref denotes the size of cells of Ω ref . Then, in the numerical simulations each discretization is built so that {t n } n=0,...,N is a subset of t ref n n=0,...,M . In Fig. 8, we consider P 1 finite elements (k = 1), and P 2 finite elements for the interpolation ϕ h of ϕ (l = 2). We compare here the l 2 (H 1 ), l ∞ (L 2 ) relative errors between the solution of the ϕ-FEM scheme (4) and a standard FEM. The numerical results fit well the theoretical convergence order announced in Theorem 1, namely, order one for the l 2 (H 1 ) norm and order two for the l ∞ (L 2 ) error.

Conclusion

In the present work, we proposed a FEM scheme following the ϕ-FEM paradigm to approximate the solution of the heat equation and proved its convergence, which is optimal in the l 2 (0, T ; H 1 (Ω)) norm and quasi-optimal in the l ∞ (0, T ; L 2 (Ω)) norm. We remark that, in comparison with [START_REF] Duprez | ϕ-FEM: a finite element method on domains defined by levelsets[END_REF], we need less regularity on the exact solution in the a priori error estimates.

A first advantage of the ϕ-FEM paradigm is its ease of implementation. Indeed, it uses standard shape functions contrary to the XFEM approach. Moreover, it uses standard integration tools contrary to cutFEM needing an integration on the real boundary and some integrations on cut cells.

A second interesting aspect of our approach is the computational time of the simulation. The low cost (computational time) of ϕ-FEM can be explained by the fact that the boundary of the geometry in the classical finite element method is approximated by some linear functions while, in the ϕ-FEM paradigm, the boundary is taken into account thanks to the level set function ϕ, which can be of high degree without increasing the size of the finite element matrix.

In the mathematical analysis, we supposed that the boundary of the considered domain is regular enough. The case of less regular domains will be the aim of future work.

Assumption 1 .

 1 The boundary Γ can be covered by open sets O i , i = 1, . . . , I on which ones we can introduce local coordinates ξ 1 , . . . , ξ d with ξ d = ϕ and such that, up to order k + 1, all the partial derivatives ∂ α ξ i /∂x α and ∂x α /∂ α ξ i are bounded by a constant C 0 > 0. Thus, on O, ϕ is of class C k+1 and there exists m > 0 such that on O \ ∪ i=1,...,I O i , |ϕ| ⩾ m.

Figure 1 :

 1 Figure 1: Left: considered domain for the first test case. Center: a conforming mesh for the standard FEM. Right: a uniform Cartesian mesh for ϕ-FEM.

Figure 4 :

 4 Figure 4: First test case. l 2 (0, T ; H 1 (Ω)) with ∆t = h (left) and l ∞ (0, T ; L 2 (Ω)) with ∆t = h 2 (right) relative errors with respect to the computation time. Standard FEM (red squares) and ϕ-FEM (blue dots), P 1 elements, σ = 1.

Figure 5 :

 5 Figure5: First test case. l 2 (0, T ; H 1 (Ω)) relative errors with respect to σ for different mesh sizes, with ∆t = h (left) and l ∞ (0, T ; L 2 (Ω)) relative errors with respect to σ, ∆t = h 2 (right), both with P 1 elements.

Figure 6 :

 6 Figure 6: First test case. l 2 (0, T ; H 1 (Ω)) relative errors with respect to h for different values of l, ∆t = h (left) and l ∞ (0, T ; L 2 (Ω)) relative errors with respect to h for different values of l, ∆t = h 2 (right), both with P 1 elements and σ = 1.

Figure 7 :Figure 8 :

 78 Figure 7: Left: considered domain for the second test case. Center: a conforming mesh for the standard FEM. Right: a uniform Cartesian mesh for ϕ-FEM.

The experiments are executed on a laptop equipped with an Intel Core i7-12700H CPU and 32Gb of memory. Moreover, for the first test case, we use the serial default solver of FEniCS. For the second test case, the GMRES linear solver is used with hypre amg as preconditioner.

Funding

This work was supported by the Agence Nationale de la Recherche, Project PhiFEM, under grant ANR-22-CE46-0003-01.