Canan Turan

Professor Akbar Javadi

Dr Raffaele Vinai

Professor Olivier Cuisinier

Professor Giacomo Russo

Professor Nilo Cesar Consoli

MECHANICAL PROPERTIES OF CALCAREOUS FLY ASH STABILIZED SOIL

Keywords: Class C fly ash, Clay, Soil stabilization, Laboratory tests

Construction of any type of structure on clayey ground could be problematic due to the high swelling potential and/or low strength characteristics of the soil. This can lead to low stability or differential settlement of the ground. Many soil stabilization techniques have been proposed to prevent the uneven settlement and failure of the soil. Stabilization of soil with class C fly ash offers many advantages such as improving engineering characteristics, being cost-effective and being environmentally friendly. Class C fly ash chemically reacts with clay which results in a more durable and stronger soil. It has been shown by various researchers that fly ash-stabilized soil is typically stiff and strong even though there is no available standard or guidelines for the use of fly ash in construction industry. This paper presents the results from a program of experimental research on stabilization of a fine-grained soil with fly ash. Laboratory experiments, including Atterberg limits, compaction, uniaxial, and consolidation tests, were conducted on samples of a clay soil with different percentages of fly ash. The results show that adding fly ash decreased the plasticity index, increased compressive strength, and decreased the swelling and compressibility index. The maximum dry density increased and optimum moisture content decreased with addition of over 5 % fly ash by dry weight of the soil.

INTRODUCTION

Fine-grained soils are generally weak and their stability is low under heavy loading [START_REF] Ramaji | A review on the soil stabilization using low-cost methods[END_REF]. Traditionally, this type of soil is considered too incompetent for construction and in many cases is entirely replaced with soil with better properties. However, this solution is not only expensive and time-consuming but it is also a difficult task. Many soil stabilization techniques have been proposed to improve the characteristics of fine-grained soils. These techniques can be categorized into the 3 groups; mechanical stabilization, chemical stabilization, and geosynthetic stabilization [START_REF] Lakhanpal | A brief review on various methods and materials used for stabilization of soil[END_REF].

Several mechanical stabilization techniques have been proposed to improve the properties of fine-grained soils, especially problematic clayey soils. However, it has been found that there is also need to alter the physicochemical characteristics of clay to provide permanent stabilization [START_REF] Petry | Review of stabilization of clays and expansive soils in pavements and lightly loaded structures -History, Practice, and Future[END_REF]. Chemical stabilization involves the utilization of chemical binders to improve the properties of soils. Chemical stabilization is an effective method due to its ease of adaptability [START_REF] Prabakar | Influence of fly ash on strength behavior of typical soils[END_REF]. After the chemical reaction of soil particles and the chemical additives, strong bond is created between soil particles, improving the soil structure [START_REF] Mahvash | Effect of fly ash on the bearing capacity of stabilised fine sand[END_REF].

The most commonly used chemical binders in soil stabilization are lime and cement [START_REF] Sharma | Performance appraisal of coal ash stabilized rammed earth[END_REF]. Many waste materials are also used as the alternative materials for the stabilization such as fly ash, bottom ash, and rice husk ash, owing to the many environmental benefits [START_REF] Petry | Review of stabilization of clays and expansive soils in pavements and lightly loaded structures -History, Practice, and Future[END_REF].

Fly ash is a waste material obtained from burning of coal and generated from thermal power plants [START_REF] Dahale | Engineering Behavior of Remolded Expansive Soil with Lime and Flyash[END_REF]. The use of fly ash for soil stabilization is encouraged by many researchers based on understanding of its effective characteristics such as pozzolanic reactions, ease of adaptability and economic viability and many other benefits [START_REF] Sharma | Stabilization of a Clayey Soil with Fly Ash and Lime: A Micro Level Investigation[END_REF]. Fly ash has been generally utilized in subbase or base layers of road construction, railway or highway embankments, and as an aggregate or mineral filler in the construction industry [START_REF] Ahmaruzzaman | A review on the utilization of fly ash[END_REF]. The advantages of using fly ash are: • The amount of coal fly ash in the world is estimated at approximately 500 million tonnes annually. At present, a large amount of ash is still discarded into landfills [START_REF] Ahmaruzzaman | A review on the utilization of fly ash[END_REF]. Thus, the use of fly ash in geotechnical projects can decrease the disposal costs of fly ash. • The disposal of considerable amounts of fly ash creates huge environmental problems, such as, air, surface water and ground water pollution due to the leaching of its heavy metals to underground water, surface and deep soil. In addition, fly ash disposal in ponds or sea can create significant damage to aquatic life [START_REF] Nawaz | Disposal and utilization of fly ash to protect the environment[END_REF]. • The use of fly ash in soil stabilization can help to reduce CO2 emissions. If fly ash is used for soil stabilization instead of cement, CO2 emissions could be significantly reduced [START_REF] Asokan | Coal combustion residuesenvironmental implications and recycling potentials[END_REF]. • Fly ash stabilized soil is an effective material for geotechnical applications [START_REF] Bin-Shafique | Incorporating a flyash stabilised layer into pavement design[END_REF].

From the geotechnical point of view, the advantages of using fly ash in construction industry are: • Significant decrease in swelling-shrinkage and plasticity potential of high plasticity soil [START_REF] Bin-Shafique | The longterm performance of two fly ash stabilized fine-grained soil subbases[END_REF]. • Improvement in compressive and tensile strength, and bearing capacity of fine-grained soils [START_REF] Dahale | Engineering Behavior of Remolded Expansive Soil with Lime and Flyash[END_REF]. • Better performance under 'wet and dry cycles' based on durability tests [START_REF] Nawaz | Disposal and utilization of fly ash to protect the environment[END_REF].

• Decrease of permeability and improvement of transfer properties of soils [START_REF] Ahmaruzzaman | A review on the utilization of fly ash[END_REF].

In this paper, the effects of fly ash in improving the mechanical properties of a clayey soil is investigated through a program of laboratory tests. The liquid limit and plastic limit, compaction characteristics, and strength, and swelling behaviour are presented for different fly ash contents.

LITERATURE REVIEW

Laboratory and field studies have been carried out by many researchers to improve the understanding of the effects of fly ash in improving the properties of soils.

Phani Kumar and Sharma [START_REF] Phani Kumar | Effect of fly ash on engineering properties of expansive soils[END_REF] assessed plasticity, strength, swelling potential, hydraulic conductivity, and compaction characteristics of a clayey soil stabilized with 0 %, 5 %, 10 %, 15 %, and 20 % fly ash. It was revealed that both plasticity index and swelling characteristics of the stabilized soil decreased by about 50 % with addition of 20 % of fly ash. With fly ash content beyond 20 %, there was no significant decrease in swelling potential. In addition, undrained shear strength increased by approximately 27 % with inclusion of 20 % fly ash. Based on compaction test results, it was concluded that the optimum moisture content decreased by about 25 % and the maximum dry unit weight increased by about 5 % with 20 % fly ash inclusion. The results from variable-head permeability tests revealed that the hydraulic conductivity decreased with the inclusion of fly ash. Cokca [START_REF] Çokça | Closure to "Use of Class C Fly Ashes for the Stabilization of an Expansive Soil" by Erdal Çokça[END_REF] studied the effects of 3 stabilizers, including high calcium and low calcium class C fly ash, lime, and cement, on the swelling potential of an expansive soil. The amounts of lime and cement used were at 0 -8 % while amount of fly ash used was between 0 -25 %. Both fly ashes were cured for 7 and 28 days. It was concluded that the plasticity index and swelling potential of all the stabilized soils decreased significantly. Furthermore, there was a remarkable decrease in swelling potential of fly ash stabilized soil from 7 to 28 days curing times. Prabakar et al [START_REF] Prabakar | Influence of fly ash on strength behavior of typical soils[END_REF] conducted a number of experiments with addition of different percentages of fly ash from 9 % to 46 % on different soil types. Compaction, shear strength, CBR, and swelling characteristics were evaluated on CL (inorganic clay with low plasticity), OL (organic soil with low plasticity), and MH (inorganic silt with high plasticity) soils. Based on the experimental results, dry density of all soil types reduced by between 15 % and 20 % by addition of the fly ash. The shear strength of the soils increased non-linearly with the addition of fly ash. Another important finding was that the swelling behaviour of the soils decreased due to the particle size, shape, and non-expansive structures of fly ash. Moreover, the CBR values of the soils improved with the addition of fly ash in comparison with the pure soil. Brooks [START_REF] Brooks | Soil stabilization with fly ash and rice husk ash[END_REF] investigated the impacts of class C fly ash and rice husk ash on high plasticity clayey soil by conducting unconfined compressive strength (UCS), California bearing ratio, compaction, and swell-shrinkage tests. According to the UCS tests results, the failure stress increased about 106 % with the addition of 25 % fly ash, although the stress decreased with adding over 25 % of fly ash. The swelling potential of the stabilized soil also decreased in consolidation tests. It was recommended that 12 % of rice husk ash or 25 % of fly ash are the best ratios for stabilization of the clayey soil.

In general, there is some evidence from laboratory and field tests indicating that the use of fly ash can be effective in stabilization of soils. The main benefits of using fly ash for soil stabilization include improvement in compressibility and strength, and decrease in the swelling behaviour.

This paper presents some results from a comprehensive program of experiments to study the effects of fly ash and alkali activators in improving the mechanical properties of a clay soil.

The experiments include Atterberg limits, compaction, uniaxial compressive strength, and consolidation/swelling tests on samples stabilized with different fly ash contents. In what follows the results from some of the experiments are presented and discussed. This is an ongoing project and more results will be available and presented at the conference.

MATERIALS AND METHODS

Soil

The soil used in this study was kaolinite, China clay. Its properties are presented in Table 1.

Scanning electron microscopy (SEM) analysis was also carried out on pure kaolinite with different magnification factors (Figure 1). The results show that the kaolinite has a flat structure. The fly ash used in this study was obtained from the MUEG Company in Germany. Its chemical composition is presented in table 2. The fly ash has 32.4 % CaO (high calcium content) and is classified as a class C fly ash according to ASTMC 618 standard.

SEM analysis was carried out on the pure fly ash with different magnification factors (Figure 2). It is seen that the morphology of fly ash is spherical. 0.9 0.5 1.5

Figure 2 SEM images of pure fly ash with 50 µm and 10 µm magnification factors

EXPERIMENTAL PROGRAM

The program of experiments included Atterberg limit tests, compaction tests, unconfined compressive tests, and consolidation tests.

Atterberg Limit Tests

Atterberg limit tests were conducted on fly ash stabilized soil with 0 % to 30 % fly ash to determine the liquid limit, plastic limit, plasticity index of the soil based on the British standard (BS 1377-2).

Compaction Tests

Standard Proctor (compaction) tests were carried out to determine the dry density and moisture content relationships of the fly ash stabilized soil. Compaction tests (BS 1377-4) were applied using 2.5 kg rammer and 1 L compaction mould. For the tests, dried kaolinite powder (China clay) was mixed with different percentages of fly ash binder considering dry weight of the soil as the first step. After that, water was added to obtain a homogenous mixture which was then left for one day in a sealed bag to allow for the uniform absorption of water into the soil.

Compaction tests were conducted with different percentages of fly ash; Control sample (pure kaolinite), 5 % of fly ash + soil, 10 % of fly ash + soil, 15 % of fly ash + soil, 20 % of fly ash + soil, 25 % of fly ash + soil, and 30 % of fly ash + soil.

Unconfined Compressive Tests

For the strength tests, specimens of the fly ash stabilized soil were prepared by static compaction in layers in an Instron 3382 Floor Model Universal Testing System which provides a constant loading up to a specified load. The prepared specimens were 50 mm in diameter and 100 mm high. The samples were also sealed and cured for 1 day.

Consolidation Tests

One-dimensional consolidation tests were conducted in oedometer to evaluate swelling and compressibility indices of the fly ash stabilized soil with 0 %, 15 %, and 25 % fly ash and with 1 day curing time. The volume change of the sample against time was recorded during the oedometer tests. The results showed that there was no considerable volume change after 24 hours. Thus, each stage of the consolidation test was conducted for 24 hours. The loading was sequenced in the order of 100 kPa, 200 kPa, 400 kPa, 800 kPa, 400 kPa, 100 kPa. The loading stage was done to determine the compression index, while the unloading stage was conducted to determine the swelling index.

PRELIMINARY RESULTS

Atterberg Limit Tests

The results of plastic limit and liquid limit tests are presented in Figure 3. Plasticity index was calculated as the difference between liquid limit and plastic limit. The value of plasticity index is a critical indicator of the swelling characteristics of the soil; the lower the plasticity index value, the less is the swelling potential [START_REF] Seyrek | Engineering behaviour of clay soils stabilized with class C and class F fly ashes[END_REF]. It is seen that the plasticity index of clay (and the potential of swelling) decreased with the addition of fly ash. In this way, it could be said that the potential of swelling also decreased. According to the plasticity chart, the classification of the soil changed from CH to ML with addition of 25 % fly ash. Table 3 and Figure 4 demonstrate optimum moisture content and maximum dry density relationships for the soil stabilized with different fly ash contents. It is seen that the maximum dry density initially decreased at 5 %. The initial decrease may be caused due to the difference of the particle density between kaolinite and fly ash. However, there was a gradual increase with addition of fly ash, from 5 % to 30 %. This may be due to the fly ash particles starting to fill the voids of kaolinite resulting in higher density. Nonetheless, the mixtures, up to 15 % of fly ash, still had a lower density than pure kaolinite. The optimum moisture content increased with 5 % of fly ash, thereafter, it generally decreased with further increase in fly ash content. This could be due to chemical reactions starting with enough fly ash that leads to excess heat and loss of water.

Figure 4 Compaction curves of the fly ash stabilized soil The effects of fly ash on improving the strength of the stabilized soil with different percentages of fly ash are shown in Figures 5 and6 The results of the UCS test demonstrate that class C fly ash improved the strength of the soil.

The fly ash was found to be an effective stabilizing material owing to its pozzolanic and cementitious characteristics. Stabilization of the soil with 15 % and 25 % fly ash showed a decrease in compressibility, indicating lower rate of settlement in the soil compared to the unstabilized soil (Figure 7). The reason of decrease in Cc and Cs values could be that the fly ash and soil particles compact well due to the morphology of the soil and fly ash.

Figure 3

 3 Figure 3 Consistency limits of fly ash stabilized soil

  . The results of the UCS tests show a gradual increase in peak compressive stress of 226, 237, 242, 254.2, 270.7, 275, and 294.5 kPa with one day curing with addition of 0, 5, 10, 15, 20, 25, and 30 % fly ash, respectively. The values of the elastic modulus of the soil were obtained from the gradient of the linear-elastic part of stressstrain curves and they were 9.5, 16.7, 20.5, 19.1, 22.8, 22.1, and 25.1 MPa for 0, 5, 10, 15, 20, 25, and 30 % fly ash, respectively.

Figure 5 Figure 6

 56 Figure 5 Stress-strain relationships of fly ash stabilized soil

Figure 7

 7 Figure 7 Compression and swelling indices of pure clay and fly ash with 15 % and 25 %

Table 1

 1

	Kaolinite clay properties
	GEOTECHNICAL PROPERTIES OF KAOLINITE
	Cone penetrometer
	Liquid limit (%)	54
	Plastic limit (%)	27.15
	Liquidity index (%)	26.85
	Classification	CI-CH
	Specific gravity (g/mL)	2.6

Maximum dry unit weight dmax (kN/m 3 ) 1.51 Optimum moisture content wopt (%) 21 Figure 1 SEM images of pure kaolinite with 50 µm and 10 µm magnification factors

Table 2

 2

		Fly ash chemical compositions
	PARAMETERS AVERAGE MINIMUM MAXIMUM
	SiO2 (%)	28.3	16.3	40.4
	CaO (%)	32.4	23.7	43.2
	Fe2O3 (%)	6.6	3.7	10.8
	Al2O3 (%)	15.8	8.2	22.5
	K2O (%)	0.5	0.3	0.9
	MgO (%)	4.2	2.5	6.6
	Na2O (%)	0.31	0.19	0.76
	P2O5 (%)	0.72	0.38	1.26
	SO3 (%)	8.6	5	13.2
	TiO2 (%)

Table 3

 3 Compaction properties of the fly ash stabilized soil

	Properties	0%	5%	10%	15%	20%	25%	30%
	Optimum	moisture
	content (%)	21	25.5	24	23.5	22.5	19.8	21.5
	Maximum dry density
	(g/cm 3 )	1.51	1.464 1.47	1.502 1.512 1.513 1.518
	Unconfined Compressive Strength Tests

Acknowledgements

The authors would like to thank the Turkish government for their financial support. This project has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 778120.

CONCLUSIONS

In this paper, the effects of consistency, swelling and strength characteristics of class C fly ash with one day curing period were determined. The following conclusions can be drawn from the results:

The values of the plasticity index decreased with inclusion of fly ash. The maximum dry density of the soil decreased with addition of 5 % fly ash, and thereafter increased. The optimum moisture content increased with 5 % of fly ash, and showed a decrease with further increase in the fly ash content. Unconfined compressive strength of fly ash stabilized soil with 1 day curing time increased gradually with increasing the fly ash content. Swelling and compressibility indices of fly ash stabilized soil generally decreased.

These findings show that fly ash would be an effective method for decreasing swelling potential and increasing strength parameters of soils.

The results presented here are part of a large program of research on the effects of fly ash and alkali activators on the mechanical behaviour of fine grained soils.


Walther Carballo-Hernández
email: walther.carballohernandez@uca.fr

Franc ¸ois Berry
email: francois.berry@uca.fr

Maxime Pelcat
email: mpelcat@insa-rennes.fr

Miguel Arias-Estrada

PhD Forum: Towards Embedded Heterogeneous FPGA-GPU Smart Camera Architectures for CNN Inference

Keywords: Processor Architectures [Other architectures styles]: [Neural nets], Processor Architectures [Other architectures styles]: [Heterogeneous (hybrid) systems], Embedded and Cyber-Physical Systems [Embedded systems]: [Embedded hardware], Integrated Circuits [Reconfigurable logic and FPGAs]: [Hardware accelerators], Architectures [Parallel architectures]: [Single instruction, multiple data] Artificial Neural Networks (ANN), Deep Learning (DL), Convolutional Neural Networks (CNN), Field Programmable Gate Array (FPGA), Graphic Processing Unit (GPU), Processing Elements Heterogeneous Computing, Edge Computing, Internet of Things, Parallel Programming, Single Instruction Multiple Data, Pipelining, Models of Computation and Architecture

The success of Deep Learning (DL) algorithms in computer vision tasks have created an on-going demand of dedicated hardware architectures that could keep up with the their required computation and memory complexities. This task is particularly challenging when embedded smart camera platforms have constrained resources such as power consumption, Processing Element (PE) and communication. This article describes a heterogeneous system embedding an FPGA and a GPU for executing CNN inference for computer vision applications. The built system addresses some challenges of embedded CNN such as task and data partitioning, and workload balancing. The selected heterogeneous platform embeds an Nvidia® Jetson TX2 for the CPU-GPU side and an Intel Altera® Cyclone10GX for the FPGA side interconnected by PCIe Gen2 with a MIPI-CSI camera for prototyping. This test environment will be used as a support for future work on a methodology for optimized model partitioning.

Introduction

Deep Learning techniques have become in the last decade the defacto choice for multiple domains, achieving a performance similar to that of a human or even outperforming it in popular and well known competitions. During this period of tremendous evolution with many ground-breaking modifications, it has been observed high accuracy in tasks such as classification, object tracking, feature selection or detection, segmentation, input generation or input reconstruction in multiple domains like: natural language processing and in vision domains, such as image processing and video analytics. Heterogeneous computing comprises sequential and parallel mapping of a certain application or subtask to the best available individual device on a system. It offers the optimal solution to these algorithmic applications given an evaluation function. These individual set of devices usually individually referred as Processing Elements (PEs) or nodes make it possible to specialize processors to the target application. Such capacity is at the heart of recent gains in terms of system energy efficiency by exploiting applications properties (e.g. data or compute intensiveness) to occupy processing facilities at their maximum. Based on the differences between systems architectures, the level and form of heterogeneity can be formalized to describe a device and its processing capabilities. What complicates the creation of such a formalization is that while each PE has individual resources, they never work in complete isolation from one another since some sort of interconnection establishes communication between them. The main types of embedded processing elements available today are: CPUs, GPUs, FP-GAs or ASICs. Contrary to CPUs-GPUs and CPUs-FPGAs couplings flourishing in products, the couple GPU-FPGA has not been much tested for its capacity to solve computational problems.

Related work

Heterogeneous computing has been a well researched domain for several decades where multiples non-von Neumann architectures started to aggregated and to exhibit great results specially on parallel programmable tasks [START_REF] Zahran | Heterogeneous Computing: Here to Stay[END_REF]. However, like any novel hardware architecture proposal, heterogeneous platforms must overcome multiple challenges. One of the most known is the speed of interchip communication channels typically orders of magnitude lower than internal data accesses. Bittner et al. [START_REF] Bittner | Direct GPU/FPGA communication Via PCI express[END_REF] and Thoma et al. [START_REF] Thoma | FPGA-GPU communicating through PCIe[END_REF] address this challenge by measuring throughput and latency as computation-to-communication ratio and establishing a direct communication between devices. In the area of embedded GPU-FPGA heterogeneous systems, Mohammad et al. [START_REF] Hosseinabady | Heterogeneous FFPGA+GPU Embedded Systems: Challenges and Opportunities[END_REF] use a similar architecture with an Nvidia Jetson TX1 SoM with a Xilinx ZCU102 tested on multiple task such as histogram algorithm, dense and sparse matrix-vector multiplication, but not tested on deep learning tasks. Yuexuan et al. [START_REF] Tu | A Power Efficient Neural Network Implementation on Heterogeneous FPGA and GPU Devices[END_REF] also propose an architecture, but with an Nvidia Jetson TX2 SoM coupled with a Xilinx Nexys A7-100T, using a CNN for performance testing but with a UART interconnection interface, causing an overhead in the tensor transfer.

Early testing and results

Our proposed architecture consists of an Nvidia Jetson TX2 SoM as CPU-GPU system and a Intel Cyclone10GX FPGA. As the theoretical computation of each device is known, multiple sub-tasks or sub-data partitions must be empirically evaluated individually on each device. Multiple setups for communications must be also measured. The first approach consists in evaluating the communication between the devices, since multiple feature tensors have to be continuously transmitted. Figure 1 shows the communication throughput in GB/s per tensor size in KB of examples of CNN mappings on a CPU-GPU mapping and a GPU-FPGA mapping. As previously discussed, the device communication is orders magnitude slower than internal transfer on devices on the same die, i.e. CPU-GPU. Notice that techniques, such as, zero-copy can increase throughput, since data redundancy is avoided, taking advantage on shared memory. We chose YOLOv2 [START_REF] Redmon | YOLO9000: Better, Faser, Stronger[END_REF] as an example of a large CNN workload where each layer is considered a subset of the total workload with an intrinsic set of resources and computation, usually defined as a partition. For this CNN model, as shown in Figure 2, the partitioning is not balanced since each layer has a exponentially reduced number of features when progressing along layers. This means that deeper layers could be easily handled for devices with memory constrains, but the gain in computation time should be hidden by communication time. A high computation-to-communication ratio is a necessary (but not sufficient) condition for having speedups on complex heterogeneous platforms. For increasing the computationto-communication ratio, it is necessary to create large clusters of processing and reduce their data dependencies. Figure 3 shows a setup taking advantage of shallower layers on the CPU-GPU side, since the GPU memory is not as constrained as in the FPGA.

Conclusions and future work

In this paper we have presented a GPU-FPGA architecture to be evaluated for performance and capabilities. It is built from two state-of-the-art device architectures, widely used in the DL community, and combines their advanced support of system parallelism. However, the most important challenge brought by such architecture is to efficiently partition the computation, represented with a model of computation, on the platform model of architecture. In future work, we will consider hardware-aware training and inference to consider cost parameters such as power consumption. The mathematical model of this optimization problem is still to be defined and therefore an optimization technique has to be selected to address this problem in a multi-constrained manner.

  (a) CPU-GPU throughput. (b) GPU-FPGA throughput.

Figure 1 :

 1 Figure 1: Communication throughput between devices.

Figure 2 :

 2 Figure 2: Layer-wise partition and number of parameters per layer.

Figure 3 :

 3 Figure 3: Proposed heterogeneous architecture with load partition.

Acknowledgments

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 765866


P Atrick Cour Illeau

B ' ~

Labo Ratoire

Arnie Noios D E Mat

Iq Ues Fond

REGULARITY OF MIN IMIZIN G MAPS WITH VALU ES IN 5'2 AND SOME NUMERICAL SIMULATION S

I ntroduction

Let Sl be t he unit ball B ;;(O, 1) of IR;; . Let H 1 (Sl, 5:.~) be th e set of all • u E H 1 (Sl, IR;;) with u.( X) E 5'1. a.e. where s'l. is the unit sphere of IR;; . For A ;::: 0 a nd f E L'l.(n , 5'1.) , let (0.1)

The critical points ofF>.( . , f) satisfy the following Euler-Lagrange equation

-.6.•u = 'U IVul' l. + A [.f-< • u, f > u]
Notice th at F>.. is lower semi-continuous on H 1 (Sl, 5 '1.), so that (0.3) is achieved by some u >.. which 1.lati::>fie1.l ( E >.. ,t) .

In this paper , we are interested in studying the regularity of U >.,. Recall th at in [BB] , Bethuel and Brezis h ave shown that there exists some regular function .f wit h values in IR;; such that U>..(f) = U>.. is not sm oot h on Sl. In [HZ], Hadiji and Zhou considered the same problem and they obtained that there is A 1 > 0 ~uch that for every A ;::: A1, for any function f in H 1 (Sl , 5'1. ) which is not a. 1.ltrong limit of smooth maps, ever y 'U >. is not regular in n, a nd t hey obtained t h at t here is Au > 0 such that for every 0 < /\ :::; Au , u ,x is regular in Sl p rovided t hat f satisfied some conditions. In the first par t, we prove that these conditions are not necessary and we give a nu merical value of /\ u.

Note th at it is well known (see [SUI], [START_REF] Schoen | Bo•undar-y Teg•ulaTity and the D ir•ichlet pmblem of h(tmwnic uwps[END_REF]) t h at 'U>.. i1. l 13rnooth accept a t a finite number of point1.l.

The regularity of rnini rni~ing map~ and ~orne related phenomena are ~tudied by many authors (see [BBL [B87], [B89), [BBC] , [DH], [HKL] and [HL]).

In the second part, we propose a numerical study of the problem (0.3). In order to minimize the energy, we have followed a strategy due to F . Alouges [A], developed to solve the problem ;nin ., f l\?• u(x) 1 £ dx .

uEI-1,, 0 (11,5-) Jn

The principal difficulty is the lack of convexity of the con~traint . The iterative algorithm used allow~ Uf:l to decrease the energy at each step, on the contrary of algorithms developed by others authors [CLL], [CHKLL], [DGL].

The proof of the convergence will be also given and numerical resultf:l will be presented with n = (0, 1) 3 and different functions f.

R egularity of minimizing applications with values in S£

Our main result is the following:

Theorem 1.1 Let f be any measurable function with values into S£ then we have, for every 0 ::; /\ ::; ~~every rninimi~er solution of (0 .3) is regular inn.

Proof o f the theorem 1.1: VVe f:ltart by quoting two ref:lultf:l. The firf:lt lernrna. coucerm; the behavior of •u ,>, uear each :>iugularitie::; aud require::; ::;mue uwdificatiou::; of the result of [BCL].

L e mma 1.1 Suppose that y E Q if:l a. f:lingula.rity of ll>. then, we have ll>.( ~c) '2::

±R ( 1 ~::::~1 ) as ;~: goes to y where R is a rotation of IR;; . In particular, the degree of 'U>. around each singularity is ±1.

The second is a variant of the well known rnonotonicity formula. for standard minimizing harmonic map.

L e mma 1.2 For any a E n and T < dist( a, an) we have

d ( 1 1 ' ) , \ 1 ' ) 8n ')) -- I"V• u>-1-+-. • lu•. A -Jl-+ -,\1--2:: 0 dr-T B(u,• r) T B(u,• r)
3

The proof of the~e lemrnaf:l are contained in [HZ]. Hence, the two proof~ are omitted. IV to. I :::; C 1 \ll• u>. -Pllu .

Sl

In particular , u ,x tends to a constant Po E S ' l. strongly in H 1 (Q, S ' l. ).Since u ,x is a minimizer we have fn • u,x • .f ~ 0. Thus we deduce that ( 1.4) t hen using the fact If I = 1, we have Combining this inequality and ( 1.3), we obtain JVv. ,x J :::; -

1 -T f ~ 2 -r -Jn (u,x -v,x) . f :::; 2V 3-3-)4-2r • ll.fll ' l .•
1 + v A v 2 -r 1 l ' 1.
A 81r ( ~, . ;;::;--:::.)

r ri \B(O,l -7) 3 :::; A 41f g('r) ,
where g('r) = 1 ( 1 + ..J2="T), if we assume A :=:; 1.

Let X E n such that lx I > ~ 0 Define T'x = 1 -lx I' since

B (:~:, r•x) c n \ B (O, 1 -2r-x)
we have by (1.5)

Using the morwtonicity formula (see Lemma 1.2) , we obtain for al11• < T:c

(1.6)

Then, for all x such that lxl 2: au, and we can conclude a~ in Theorem 1.1 (if N is the uniform norm, then we can choose k = 1) .

R e mark on t he e qua tion

Vl/e have seen that any solution of problem (0.3) satisfies weakly, the equation:

'2 -6.u = uiY'u l + >, [.f-< u,f > u].
Then, if we take the exterior product of (E>.,t) by u. we obtain :

(6.u +,\f) X • u = 0.
Conversely, any solution of ( E~,f ) shall be collinear to u thus such a map satisfies :

6.u + >.f = IJV•,
so taking the ~calar product of thi~ equation we find that p had to be equal to -IV' zt.l'l+ < u,f >,thus the equations (E~,J) and (E>.,t) are equivalent.

As a consequence, if ('un) is a solution of (E>-,tJ such that 'Un ----" • u

I :1 '2
weakly in H (B , S ), then • u is a solution of (E>.,t ). \1\lith the formulation (E;,/ ), we have j ust to note that by compact injection of H 1 in L 2 that •un tends strongly to •u in L 2 (B 3 , 5 2 )

then we can pass to the limit in fn x u. 11 , it is well known that .6.u. 11 x U.n tends also to -.Cw x • u (see [BBC], [C]).

R e rnark on t h e solutions

The regularity obtained in Theorem 1.1 (and T heorem 2.1) i~ really a consequence of the rninimi:tation problem and not of the equation.

Indeed, if we consider the following rninimi~ation problem on H 1 ( B 3 , 5 2 ) :

where r.p is a given smooth boundary condition. Any solution of problem (P>.) ~atisfies the equation ( E >.,! ) .

Then, for r.p equal to the identity on the boundary and f constant for example, the solutions u >. of ( P>.) converge to the solution of ( P 0 ) when .\ tends to ~ero, thus there exists singular s olutions of ( E>.,J ) for /\ small.

N ume rical minimization of t h e energy F>.

In this part, we propose a numerical stud y of the problem (0.3). The strategy used here is based on the works of F . Alouges [A] . T he principal difficulties of finding numerically the minimizer are:

• Non convexity of the constraint l• u.( x) I = 1 a.e. which avoid u s to use ~tandard algorithms directly.

• The minirni~er • u >. may be non regular (non continuous) for some .\, f .

• Non uniqueness. For some A, f (iff have symmetries for example), U.>. need not to be unique .

Most of the methods to solve this kind of problems can be split into two ~teps:

1. Let u 0 be an initial guess. 2. For n = 0 ... until convergence:

3.1 Find Vn such that F>.(vn) ::; F>. (un) where Vn may not belong to H 1 (Q, S'l.); ()

~ 3.2 Set Z/,n+l X = 1• n( •) I' v,, , X
The rninirni~ation problem we will solve at the step 3.1 allows us to decrease the energy at the step 3.2. In other words, for all iterations n, we have F,x (un+l) ::; F,x (vn) ::; F,x(u.n) • O t her method~ ( [CLL], [CHKLL], [DGL] for example) do not have thi~ property. In par t icular , F,x c~: 1 ) ::; F,x (vn) i~ not a~~ured for all iter ations n .

The ~tep 3.1 will be ~olved by a conj ugate gradien t method beca.u~e t here i~ no parameter to optim ize (on the contrar y of a saddle-point or relaxation technique for example). Moreover , the numerical tests of F . Alouges seems to prove t hat it i s the better method. So, minimize I (w) is equivalent to minimize J(w) defined by (3.4) a(w, ' l/J )= L ('l/J) for all 'ljJ E Ktl .

1 1' '2 ~ 1 J(w ) = - IVwl + --\lwl d:r: -\7u • \7• w + /\(u-f ) • w dx .

.2 C o nver gen ce o f the a lgorit hm

Now, the algorithm can be wrote a.::; follow :

1. Let u 0 be an init ial guess. 2. For n = 0 ... until convergence: (3.5) 3.1 Find Wn, solution of the problem (3.3) with • u = • un;

'Un(x) -wn(x) 3.2 Set Un+l(x) = lun(a;) _ wn(x) l' and we will prove the convergence of the algorithm by the followin g result:

T heo rem 3.1 The algorithm (3.5) converges in the sense t hat ('un) (up to a !:>ubsequence) weakly converges in H 1 (Q,lRJ ) to a m ap u 00 E H 1 (ft , S 2 ) verifYing Expanding this expression, we obtain 1 \7</ J • (-un X \l(wn-'Un)) + <P • (Vun X \lw,., + AVn X (w,.,-f) ) da; = 0, so •un, Vn satisfy the following Euler-Lagrange equation in the sense of distribution s. Using the facts that 'U.n ___,_ 'u. 00 weakly in H 1 , ' U-n ---7 'U. 00 strongly in L'l. , 'Wn ---7 0 str0ngly in H 1 , V 00 satisfies the Euler-Lagrange equation: div(u 00 x 'Vu 00 ) + >.j x 'U.oo = 0 in the sense of d istributions, which is equivalent to the fact that v 00 verifies the relation (E>.,t) (see section 2) . The di~crete ver~iun of t he Lax-Milgram t heorem allow~ u~ to ~ay that the solution w~ of (3.1) is unique and satisfies t he problem R emark: 'When a u~ is given, at each step n, the solut ion w~ is u nique, so the lim it u~ is a lso unique. But, with an another init ial guess uJ, we can obtain a different lim it u. ~ since the solution of the problem (0.3) may be nut unique.

Fur thermore, t he present algorithm m ay converge~ to a crit ical point (but n ut nece~~ary a global minirni:.::er) ofF: >. .. E xamples of t his phenomena will be given in t he numerical result~.

R esolution of the convex problem : a conjugate gradie nt technique

Here, we follow the algorithm given by F . Aluuge~ [A], based on the rem a rk bellow:

Suppu~e we want t o minimi:.::e F (X) = ~(AX, X )-(b,X) where(., .) is the inner pwduct on IRN, A is a p ositive definite N x N m atrix , b i~ a vector in IRN, a nd X E IRN , su bject to t h e constraint EX = 0.

The solution X may be obtained by applying a cunjugate gr adien t method proced ure to the functional

F(X ) = ~(1r An X , X )-(1rb, X ),
where 1r stan ds fur the ort hogonal pwjectur unto t he linear space ]( = {X E IRN such that EX = 0} provided the a lgmithrn is started with X 0 E K.

N N

In our case, since u 1 '(x) = L ' u~1<pi(x), wh(x) = L wf<pi(x), the projector i=l i=l n h unto the linear space K:h can be wrote i,j=l

N umerica l results

In t his section, numerical results a re given wi th Q = (0, 1 )' 1 and different functions f. T hat allow~ us to have an idea of t he behavior of t he energy F:>.. a nd t he solut ion l l. ).. as a fun ctio n of), a nd .f. T h ree cases will be studied :

1. f(:c) = ~~=~~~ where x u E Q . f has a t : >ingularity of degree 1 int:>ide .Q.

2. fit:> the ~tereographic projection onto (0, 0, 1) t : >hifted to the point (0.5, 0 .5, 1) . Here, f have a singularity of degree 1 on fJ.Q .

3. f is the dipole. f has two singularities into n, one of degree +1, one of degree -1 t:>uch that .f has a global degree 0 (~ee [B87]).

In that thr~~ <:a~~~, 71.>-.. is r~gnlar for r~. /\~mall ~nough r~ncl singular for a larg~ enough one. The t : >ingularitiet:> are alway~ in~ide Q (never on 8S1), of degree ±1, and locally equivalent to ±R( 1 ~::::; 1 ) where R is a rotation and y the point where the ~ingularity appear~.

R e mar k : vVe have al~o used a simulated annealing method to ~olve the problem (0.3). It give~ exactly the t:>arne re~ultt:> but with a CPU time~ much more co~tly.

That is why we have chosen to not present this algorithm here. ) , :c1 , x'2 E 0, 1 , ecause t e pomt xu where t e smgulanty u>.(xh x' 2 , 0.5 may appear is int : >ide thit : > :>ection.

J ,J(x) dx

Figure 2 ~howt:> that the ~olution U>.. tendt:> to a limit P 0 (x) = P 0 = ....:<..f,J'--- ;,; (,.

1 j~ f(x) dxl
, . ) , X 1 , a; 3 E 0 , 1 .

U A x 1 , 0.5,x 3

J j(x) dx

In figure 7, we can see that 'U A tends to ~ as /\ goes to 0 and in figure

I Jn f(x) dx l
8 that when /\ is small enough, tt.A is regular, For ).. large enough, uA becomes t>ingular (figure 9). The singularity of 'i.IA is at a Xu inside n, and of degree +1.

'\IVhen /\ increases, the sing ularity xG draw near to Dfl. but it never reach Dn.

The function f h as two s ingularities inside n

Here, the function f is t he dipole . , x1, X::; E , 1 . . as two stngu an ties, one o egree + 1, f (.rJ, 0 .5, .r;,;) one the degree -1. Figure 12 (the sectiou of 'UA represented is the same as in figure 10) shows that •uA is regular if/\ < Ac. In this example, ).. is much larger than in the previous ones, because the energy increases slowly.

Figure 13 shows that the two singularities appears at the same times, such that the solution u A has always a global d egree equal to 0.

Around the >.. e where the behavior of the solution U,\ changes, we obtain different limit according to the initializ:a.tion. The firt>t initia.liz:ation is 'Uo = f ( u 0 is singular), the second is tt. 0 = (0, 0, -1) and the thir d is u 0 = (0, 0, 1 ).

Conclusion

The algorithm developed here is very efficient and well suited to solve the rninimiz:ation problem (0 .3). The non-linear initial problem has been solved using a sequence of linear problem, much easier to treat (but each iteration requires the resolution of a. linear system ), and the rate of convergence is very good. The numerical results confirm the result of part 1, and the conjecture that the close set I (f) of).. where 'U,\ is regular is an interval of lR+ .

  Setting v.x ( x) = v >. ( ( 1 -r'):~:) then we have ( 1.1) Clnd, Since l/ >. is minimi:.::er , it full ow::write for almost everywhere x in Q 1 ll•>.(x) -u>-((1 -T)x) = 1 x • V'u>.(tx) dt 1 -r and then, for a.e. X E Q using again Poincare inequality t hen Using the cha nge of variable y = tx, and Fubini-Tone lli theorem, we obtain This lea. ds to On the other han d, we have for all P E S ' l. hence 1 ' 1.

Finally

1

  E B(O , ~) , we have B(x,!) C B 3 (0 , 1). Using again the monotonicity formula and (1.4), we see for al11• < ~ then ( 1.8) ~ r 1 Vu.xl 2 :::; 2 r 1 Vu.xl 2 + 32 /\7f( ~ -T~) Vx E B(O, 2 ).Direct computations show that for all /\ :=:; ~ the right hand sides in ( 1. 7) and(1.8), are strictly less than 87f, SO, for all X in Q limnote that the right hand side in (1.8) is strictly less than 871" for all A < 1.) Now, applying Lemma 1.1, we obtain the desired conclusion. 2. Remarks and generalizations 2.1 Remark on t he domain n VVe have a ~irnilar re~ult as Theorem 1.1 if we only as~ume that f i ~ a function in L'l(n , lR 3 ) and if we replace the domain n by any unit ball as~ocia.ted to another norm in lR 3 : Theorem 2.1 Let f be any function in L'l(B N, IR 3 ) not nece~sary with value~ into S'l defined on B N = { :c E lR 3 , N (:c) :::; 1} where N i~ a. norm in lR 3 , then there exi~t~ a con~tant Au > 0, depending only on ll.fll'l and N, s uch that every rninimi~er u >-E H 1 ( B N, S'l) of the functional F>. (., f) for /\ :::; J\ 0 ) is regular in B N . The proof i~ the ~ame as for B 3 . u~ing V,>.(x) = U>.((1 -T)x) we obtain inequalitie~ (1. 3) and (1.4) thu~ we prove that there exi~b a function G(• r) = (II Pu ll ~+ ll.fll~) + V( I IPu ll~ + l l.f l l~) • ll.fll'l • /2-T, ~uch that ~ r IV' u>-1'2 :::; /\G(r'). T J n\(1-r')fl 1 Thus for x E n such that N(x) > 2'' if we set rx = 1 -N(x) then we see that there exists a constant k which depends only on the norm N ~uch that (2.1)

  3.1 A n e n e rgy d ecr e a s ing algorithmHere, we want t o ~olve t he p roblem : " F ind Vn ~uch t hat F,x (v n ) ::; F,x (tl.n)" in order t o a~surne that :(3.1) at each step n . T his can be done using t he following proposition given by F . Alouges : Propos ition 3.1 If v E H 1 (D, IR 3 ) verifies lv(x )I > 1 a .e ., then l ~l belongs to H 1 (D, S~) . Moreover , we h ave 1 \7 ( v(;~:) ) ~~:::; l\7v(:c ) I~ v verifies lv(x) l 2: 1 a..e ., one easily have the cond it ion (3.1) . Proof of Propos ition 3 . 1: Thi~ result can be shown by d irect computation~ (See [A]). Now, the following result allows us t o min im ize F.>. wit h a function Vn verifying lv(x)l 2: 1 a .e .: Proposition 3.2 Let K u be t he set: Ku = {wE H 1 (D, IR 3 ) such that w(x) • u(x) = 0 a .e .} Let v = • uw where w belongs to K . u, F).. ( u -w) for w E K 11 possesses an unique solution, called w(u.). Proof o f Pro pos ition 3.2: The proof of (3.2) is obvious because w(x) • u.(x) = 0 a .e. For t he point (3.3), we have I (w) = F)o. (v. -w) = fn 1 Y'(u -w)l~ + --\ luw -fl~ dx . ) = JniY'wl +--\lwl dx-2 Jn \7• u•\7w +--\(u-.f}w dx + Jn IV• u. l +--\l• u.-! 1 dx .

  ~) = { Y'w • Y''I/J +>-w • ~dxand L('I/J) = { Y'u •\7~+>-(u-.f) • 'I/J dx ln ln for all '1/J E K u .Then a is clearly continuous, coercive on K 11 (because/\> 0) , a nd Lis continuous on K 11 • Moreover K 11 is a linear subspace of H 1 (ft, IR 3 ) , so we can use the Lax-Milgram theorem to prove the uniqueness of w . Furthermore, w is also the solution of the variational problem:

3 . 3

 3 DiscretizationvVe use finite elements method because we absolut ely need to have a symmetric matrix. Indeed, because of the lack of Dirichlet conditions on on, finite differences method produces a non-syrnrnetric matrix preventing us using a conjugate gradient technique.The finite elements used are linear on cubes (8 nodes) with a constant spacestep in each d irection. If we call {<pi} i= 1.. . N the set of interpolation functions, and Yh = span{ <pi; i = 1.. .JV}, a function u is approximated by : N v(:c)-:::: • uh(x) = L: • u~'•<pi(x) . i=l vVe can also define the set: \iVith all these notations, the di screti~ed algorithm can be wrote as follow : (3.6) 1. Let u3 be em initial guess. 2. For n = 0 ... until convergence: 2.1 Find 'W~ such that F>. (w~) = rnin F>.(v~w h); wh E [(h .. ~ • uh(x)w 11 (x) 2.2 Set u~+ 1 (x) = I ;1 ( ) 7 (<)I' 'U. . 1 ~ X -' Wr: X

  ) = w 11 (:~:) -(wh(:~:) • • uh(:~:)) • u 11 (x) = wh(x) -L • u~'wj (<pi(x) •<pj(x))uh(x).

4. 1

 1 The fu nction f h as one s ingula rity inside Q In this fir st example we have .f(x) = x-xo where Xu = (0.6 0.8 ) 0.5) E n.' lx-xo l 'Remark : In order to have an idea of the properties of the ~olution U.> 11 we have decided to draw only a section of two components of the solution. Here, the section { u\(:~: 1 ,x:L,0 .5) ( ) (

  as A goet:> to 0 . \iVhen A increa~e~, the ~olution 'U.><. become~ more variable but ~till regular (figure 3) while A is smaller than a certain At; . Then, when /\ > /\e, the solution V>. has a ~ingularity at :to = (0.6, 0.8 , 0.5), and ' U>.(x) ~ ~~=~~~ at:> x goe~ to xu .Notice that the singularities of U>. and fare at the same point xu .

Figure 1

 1 Figure 1 t:>how~ the energy of the minimi~ing function V>. for different valuet:> of A, obtained with two different initializations . The first initialization (init 1) is ' Un=o(x) = (0) -1, 0) (un=O i~ regular), and the second (init 2) is ' Un=O = f (un=U i~ t : >ingular). \iVhen A it : > far from Ae, the two different ~equencet:> (• un) converge to the ~arne limit U>., and we obtain the ~arne energy. On the contrary, when A i~ near from Ae, with the fir~t initiali~ation the ~equence ('un) converget:> to a regular function u\ and with the ~econd initialization, (un) converget:> to a t : >ingular function •u~. If /\ < Ae, we have E>.(u\ ) < E>.(u{ ) ~o the glob<:t.l minimizer it : > ul wherea~ if A> .X e, we have E>.(u\) > E>.(ui ) t:>O the global minirni~er i~ ui .

Figure 6

 6 Figure 6 shows t hat, as above, around a certain Ae, we obtain two different minima according to the initializ:ation, and only one of the two is the global minimum (except of course for /\ = >.. e where the two are global minima. In this case, t he problem (P) have not an unique solution).In order t o represent v A, we have chosen to plot a. section of two components { ui(a; . •J, 0.5,a;3) (•• .. ) ( )~

  Figure 10 represents the section off defined b { fl(xi,0.5 ,x::~) ( ) (0 )~ f

the equation ( E>.,J ) . Moreover, the full ~equence (wn)n2:0 ~trongly converge~ to 0

Proof: The proof is similar to these of F. Alouges (see [A]). vVe first need the lernrna: Since ,\ > 0, fn 1Y'wn l~ + -A iwnl'l dx is equivalent to the usual norm on

H (n, lR ) and Wn -----+ 0 ~trongly in H (Q, lR ).

The proof of the weakly convergence of (tt. 11 ) in H 1 (Sl ,S 2 ) is given in [A]. It i~ based on th e fact that (' un ) i~ bounded in H 1 (Sl, S 2 ) .

Finally we have to prove that the limit 'Uoo i~ a critical point ofF>. . U~ing t he variational formulation (3.4) , and taking 'lj; = ¢ x ' Un where¢ E Ccf'(Sl, lR 3 ) , we have:


Mostafa Abounouh

Hassan Moatassime

Jean-Paul Chehab

Serge Dumont

Olivier Goubet

H Al Moatassime

J.-P Chehab

Discrete Schrödinger equations and dissipative dynamical systems

Keywords: 2000 Mathematics Subject Classification. 35B41, 35Q55, 65M06

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1. Introduction. Weakly damped nonlinear Schrödinger equations provide examples of infinite-dimensional dynamical systems, in the framework described in [START_REF] Temam | Infinite Dimensional Dynamical Systems in Mechanics and Physics[END_REF], [START_REF] Hale | Asymptotic behavior of Dissipative Systems[END_REF], [START_REF] Raugel | Global attractors in partial differential equations[END_REF]. For these infinite-dimensional dynamical systems the major issues are: does it exist a global attractor for the dissipative dynamical system under consideration ? does this global attractor has finite Haussdorf and fractal dimension ? is this global attractor regular ?

Let us give an overview of the previous results for weakly damped nonlinear Schrödinger equations, that are equations that read

u t + αu + iu xx + i|u| 2 u = f. (1)
Here the unknown u(t, x) maps R t × T x into C. We mean that u is a periodic function with respect to x. Actually α > 0 is the damping parameter and the external force f , that does not depend to t, belongs to L 2 (T). The pioneering work [START_REF] Ghidaglia | Finite dimensional behavior for the weakly damped driven Schrödinger equations[END_REF] proved the existence of a finite dimensional weak global attractor A for dissipative NLS. By weak attractor we mean that the attractor attracts the trajectories for the weak topology in the Hilbert space chosen for the mathematical study. Moreover, due to a famous argument due to J.

Ball, it turns out that this weak attractor is actually a global attractor in the usual sense. Hence the two first issues were successfully addressed. For the issue of the regularity of the attractor, the first result appeared in [START_REF] Goubet | Regularity of the attractor for the weakly damped nonlinear Schrödinger equations[END_REF] and states as follows: consider the dynamical system provided by dissipative NLS in H 1 (T) with a forcing term in L 2 (T), then the global attractor is actually a compact subset of H 2 (T). In other words dissipative NLS feature an asymptotical smoothing effect, following the terminology introduced in [START_REF] Haraux | Two remarks on dissipative hyperbolic problems, Nonlinear Partial Diff[END_REF].

In this article we would like to address the same issues for discrete nonlinear Schrödinger equations. By discrete nonlinear Schrödinger equations, we mean that we discretize the equation both in time and in space and we consider the associated finitedimensional dynamical system. For discrete nonlinear Schrödinger equations some interesting results are available in the literature but mainly for discretization in space of the equations (nevertheless we refer to the work [START_REF] Bao | An explicit unconditionaly stable numerical method for solving damped nonlinear Schrödinger equation with focusing nonlinearity[END_REF] where a suitable splitting scheme was used for the damped nonlinear Schrödinger equations). We would like to refer to [START_REF] Yan | Attractors and dimensions for discretizations of a weakly damped Schrödinger equations and a sine-Gordon equation[END_REF] and [START_REF] Cuevas | Tresholds for time periodic solutions on the discrete nonlinear Schrödinger equation with saturable and power nonlinearity[END_REF] where the authors consider some discretization in space of the Laplace operator on a finite interval with finite differences and then study the dynamical systems provided by the associated ODE. On the other hand, we would like also to point out the study of discrete nonlinear Schrödinger equations as an infinite-dimensional dynamical system in a lattice in [START_REF] Karachalios | Global existence and compact attractors for the discrete Schrödinger equations[END_REF]; the authors consider the ODE in the infinite dimensional space l 2 (Z) defined at each point j ∈ Z by

(u j ) t + γu j + i(2u j -u j+1 -u j-1 ) + i|u j | 2 u j = f j . (2)
In our article we discretize the Laplace operator with finite differences on a grid of mesh size ∆x. We consider also a time discretization provided by a suitable Crank-Nicolson scheme. We prove below that this new scheme fits with the damping and is unconditionally stable even for long time. This allow us to prove the existence of a global attractor A ∆x for the discrete dynamical system under consideration. Our aim was to give some numerical evidence of the regularity of this attractor. This question is meaningless if one says that we deal with finite dimensional space and that all norms are equivalent. But this matters if we want to have estimates that are uniform in ∆x and ∆t. Introduce L 2 ∆x and H s ∆x spaces as follows. Set A for the discrete Laplace operator in finite differences. Set, for a vector

U in C N , ||U || 2 L 2 ∆x = (∆x) j |u j | 2 , (3)
and

||U || 2 H s ∆x = ||(A + Id) s/2 U || 2 L 2 ∆x . (4)

Our first result states as follows

Theorem 1. There exists A ∆x a compact global attractor for the Discrete Nonlinear Schrödinger equations in L 2 ∆x . Moreover A ∆x is a bounded subset in H 2 ∆x , uniformly in ∆t and ∆x.

The results compares with [START_REF] Goubet | Dissipative Nonlinear Schrödinger Equations and Bourgain Methods[END_REF] for the continuous case. This results is not obvious since by the inverse inequality,

||U || H 2 ∆x ≤ c(∆x) -2 ||U || L 2 ∆x , a bounded set in L 2 ∆x is not (uniformly) bounded in H 2
∆x . We give also some numerical evidence for this result. Consider an initial data that is not regular (a shock). Our numerical scheme introduces no artificial viscosity, and for short interval of times, the solution remains as regular as the shock. But, for the long range, some smoothing occurs.

After the stability issues, come the consistency. Consider the ODE provided by (2) on a finite number of js, corresponding to the discretization of (1) in finite differences. One can prove that this ODE features a global attractor A ODE (see [START_REF] Yan | Attractors and dimensions for discretizations of a weakly damped Schrödinger equations and a sine-Gordon equation[END_REF]). We prove in the sequel that for a trajectory in this attractor A ODE , our scheme is of order 2 in time, uniformly in ∆x. We prove also the upper continuity of the attractor, that is if both ∆t and ∆x converge to 0 then A ∆x converges to A, the continuous attractor. On open question is to estimate the distance between these two attractors. We cannot conclude since we do not now the speed of convergence of trajectories towards A.

For this issue and the issue of lower semi-continuity of attractors, one can conclude only if we now that we have a gradient system and if we know that the continuous flow satisfies the following property: the stationary points, that are in finite numbers, are hyperbolic (see [START_REF] Raugel | Global attractors in partial differential equations[END_REF] and the references therein). To our knowledge, the precise comparison between a continuous flow and its time discretization requires this kind of property; we would like to refer to [START_REF] Alouges | On the qualitative behavior of the orbits of a parabolic partial differential equation and its discretization in the neighborhood of a hyperbolic fixed point[END_REF], [START_REF] Alouges | On the discretization of a partial differential equation in the neighborhood of a periodic orbit[END_REF] where the authors compare the flow and its time discretization in respectively a neighborhood of an hyperbolic fixed point and an hyperbolic periodic orbit. The hyperbolicity properties mean that the stable and unstable manifolds at each point of the invariant set under consideration intersect transversally. This applies for instance for Ginzburg-Landau equation, that is a parabolic regularization of Schrödinger equation. For the sake of completeness, we would like also to point out [START_REF] Lord | Attractors and inertial manifolds for finite-difference approximations of the complex Ginzburg-Landau equation[END_REF] where the authors gives a complete study of the discretized Ginzburg-Landau equation.

This article is organized as follows. In the second section, we introduce the time discretization and we prove Theorem 1. In a third section we discuss the upper continuity and the consistency properties. In a last section we perform some numerics.

2. The discrete Schrödinger equation.

2.1.

A new scheme. Consider the ODE provided by a finite difference approximation (in space) of the periodic NLS equation [START_REF] Alouges | On the qualitative behavior of the orbits of a parabolic partial differential equation and its discretization in the neighborhood of a hyperbolic fixed point[END_REF]. Introducing

A = 1 ∆x 2           2 -1 0 . . . 0 -1 - 1
          this ODE reads U t + αU -iAU + i|U | 2 U = F (5)
where (|U | 2 U ) j = |U j | 2 U j , U ∈ C N and where F stands for a finite difference approximation of f . We assume in the following that ||F || L 2 ∆x is bounded uniformly in ∆x. At a first glance, the classical time discretization of (5) could be

U n+1 -U n ∆t + α U n+1 + U n 2 -i A U n+1 + U n 2 + i 4 |U n+1 | 2 + |U n | 2 U n+1 + U n = F. (6)
In this article, we rather introduce a new scheme ; to begin with, let us observe that ( 5) is equivalent to

d dt e αt U -iA e αt U + i|U | 2 e αt U = e αt F. (7)
We discretize this equation by an Euler scheme of order two to obtain (after multiplication by e -α(n+1)∆t

U n+1 -e -α∆t U n ∆t -iA U n+1 + e -α∆t U n 2 + i 4 |U n+1 | 2 + |U n | 2 (U n+1 + e -α∆t U n ) = 1 + e -α∆t 2 F. (8)
We would like to point out that if α∆t << 1, then since e -α∆t ⋍ 1 -α∆t, (8) is equal to (6) up to second order.

2.2. Uniform stability: L 2 ∆x estimate. We now prove that this new scheme is stable in L 2 ∆x , uniformly in ∆x and time (discrete time n∆t), assuming that α∆t << 1.

Proposition 1. Assume that α∆t is small enough. There exists δ = e -α∆t < 1 such that

U n 2 L 2 ∆x ≤ δ n U 0 2 L 2 ∆x + (1 -δ n ) 8 α 2 F 2 L 2 ∆x (9)
Proof :

Consider the scalar product of (8) with U n+1 + e -α∆t U n in the real Hilbert space C N ≃ R 2N (endowed with the scalar product in L 2 ∆x ) to obtain

U n+1 2 L 2 ∆x ≤ e -2α∆t U n 2 L 2 ∆x + ∆t| < F, U n+1 + e -α∆t U n > |, (10)
where < F, G >= (∆x) j Re(F j G j ). Therefore, by Young's inequality

1 - α∆t 4 U n+1 2 L 2 ∆x ≤ e -2α∆t 1 + α∆t 4 U n 2 L 2 ∆x + 2∆t α F 2 L 2 ∆x (11)
We observe that for α∆t small enough we have δ 1 -α∆t 4

-1 1 + α∆t 4 ≤ 1.
We then obtain

U n+1 2 L 2 ∆x ≤ δ U n 2 L 2 ∆x + 4∆t α F 2 L 2 ∆x ( 12
)
The discrete Gronwall's lemma gives then

U n 2 L 2 ∆x ≤ δ n U 0 2 L 2 ∆x + (1 -δ n ) 4α∆t α 2 (e α∆t -1) F 2 L 2 ∆x , (13)
that completes the proof of the Proposition.

2.3.

Well-posedeness of scheme. (8) defines an implicit scheme. This problem is well-posed if we are able to prove that the Picard iteration scheme w k → w k+1 on C N defined by

w k+1 -e -α∆t U n ∆t - iA 2 w k+1 + e -α∆t U n + i 4 |w k | 2 + |U n | 2 (w k+1 + e -α∆t U n ) = 1 + e -α∆t 2 F (14)
with w 0 = 0 is convergent. It is an exercise (proceeding as in section 2.2) to prove that

sup k w k 2 L 2 ∆x ≤ 8 α 2 F 2 L 2 ∆x . ( 15
)
On the one hand, estimates [START_REF] Lord | Discrete Gevrey regularity, attractors and upper semi-continuity for a finite difference approximation of Ginzburg-Landau equation[END_REF] ensures that the non linear mapping

w k → w k+1 , that is continuous, send the ball of radius 2 √ 2 α F L 2
∆x into itself. The Brouwer fixed point theorem applies and then leads to the existence of a fixed point U n+1 . On the other hand, if one prefers to establish the uniqueness of the fixed point, we rather use the following statement, assuming some kind of CFL condition on ∆t ∆x .

Lemma 1. : Assume that ∆t ∆x is small enough. Then the mapping w k → w k+1 is a contraction and the Banach fixed point theorem applies. Then we have the existence and uniqueness of a fixed point U n+1 . Proof : Consider ( 14) at step k -1 and subtract this equation to ( 14) at step k. We thus obtain the following equation for

Z k = w k+1 -w k Z k -i∆t 2 AZ k + i∆t 4 (|U n | 2 )Z k + i∆t 4 (|w k | 2 -|w k-1 | 2 )U n e -α∆t + i∆t 4 (|w k | 2 Z k + (|w k | 2 -|w k-1 | 2 )w k ) = 0. ( 16
)
Let us now multiply this equation by Z k in the real Hilbert space L 2 ∆x and to obtain

||Z k || 2 L 2 ∆x = ∆t 4 < w k Z k-1 U n e -α∆t , iZ k > + ∆t 4 < w k-1 Z k-1 U n e -α∆t , iZ k > + ∆t 4 < w k Z k-1 w k , iZ k > + ∆t 4 < w k-1 Z k-1 w k , iZ k > (17)
and then, by straightforward inequalities

||Z k || 2 L 2 ∆x ≤ (∆t||Z k || L 2 ∆x ||Z k-1 || L 2 ∆x ) . ||w k-1 || L ∞ ∆x + ||w k || L ∞ ∆x ||U n || L ∞ ∆x + ||w k || L ∞ ∆x + ||w k-1 || L ∞ ∆x ( 18
)
We now recall the following well-known inverse inequality

Lemma 2. : ||f || L ∞ ∆x ≤ 1 (∆x) 1 2 ||f || L 2
∆x . Then, the uniform upper bound for w k , U n on L 2 ∆x leads to : there exists c > 0 such that

||Z k || L 2 ∆x ≤ c ∆t ∆x ||Z k-1 || L 2 ∆x ( 19
)
The results follows promptly.

2.4. Existence for the discrete global attractor. At this stage we have defined a nonlinear mapping S : U n → U n+1 that maps continuously L 2 ∆x ≃ C N into itself and that possesses a bounded absorbing set in L 2 ∆x . Since we are dealing with finite dimensional space, we have that the continuous map S is compact, then Theorem I.1.1 in [START_REF] Temam | Infinite Dimensional Dynamical Systems in Mechanics and Physics[END_REF] applies and we obtain Proposition 2. There exists A ∆x a compact set in L 2 ∆x that is a global attractor for the dynamical system defined by the Discrete Nonlinear Schrödinger Equation.

At this point, we would like to emphasize the following point : the "size" of the global attractor does not depend on the mesh size of the space discretization ∆x. This contrasts with the following fact : when we deals with the classical scheme [START_REF] Ghidaglia | Finite dimensional behavior for the weakly damped driven Schrödinger equations[END_REF], one can prove the existence of an absorbing ball whose size tends to infinity when ∆x → 0.

2.5. Discrete regularity for the attractor. We first state and prove

Proposition 3. : A ∆x is a bounded subset in H 1 ∆x , uniformly in ∆x. Proof : Multiply (8) by i(U n+1 -e -α∆t U n ) in the Hilbert space L 2 ∆x to obtain A 1 2 U n+1 2 L 2 ∆x -e -2α∆t A 1 2 U n 2 L 2 ∆x = 1 2 < (|U n | 2 + |U n+1 | 2 )(U n+1 + e -α∆t U n ), U n+1 -e -α∆t U n > + (1 + e -α∆t ) < iF, U n+1 -e -α∆t U n > . (20)
Let us observe that the first therm in the r.h.s. of (20) reads also

N j=1 ∆x |U n+1 j | 4 -e -2α∆t |U n j | 4 + (1 -δ 2 ) N j=1 ∆x(|U n j | 2 |U n+1 j | 2 ). (21)
Introducing

J(U n+1 ) = A 1 2 U n+1 2 L 2 ∆x - 1 2 U n+1 4 L 4 ∆x + (1 + δ) < iF, U n+1 >, (22)
We thus obtain

J(U n+1 ) ≤ δ 2 J(U n ) + 1-δ 2 2 N j=1 ∆x(|U n j | 2 |U n+1 j | 2 ). ( 23
)
We now use the following discrete Gagliardo-Niremberg inequality :

Lemma 3. : ∀v ∈ C N , v 4 L 4 ∆x ≤ 2 v 3 L 2 ∆x v H 1
∆x . Now and in the sequel we denote by K a constant that depends only on α and F L 2 ∆x ; K may vary from one line to one another. We infer from the uniform upper bound in L 2 ∆x and from (22) that there exists K such that

J(U n+1 ) ≥ 1 2 A 1 2 U n+1 2 L 2 ∆x -K. ( 24
)
On the other hand, there exists (another) K such that

(1 -δ 2 ) N j=1 ∆x(|U n j | 2 |U n+1 j | 2 ) ≤ K∆t(1 + A 1 2 U n+1 1/2 L 2 ∆x A 1 2 U n 1/2 L 2 ∆x ). ( 25
)
We now infer from ( 23)-(25) that

J(U n+1 ) ≤ δJ(U n ) + K∆t. (26)
Then, proceeding as in the proof of Proposition 1 (see ( 9)), we obtain

J(U n ) ≤ δ n J(U 0 ) + K∆t 1 -δ ≤ δ n J(U 0 ) + 2K α . ( 27
)
We now consider a discrete trajectory U n that belongs to the global attractor A ∆x that is bounded in L 2 ∆x . Let us recall that the global attractor consists in complete orbits and that therefore we can go backward in time. There exists then K that depends only on α, F L 2 ∆x such that, due to inverse inequality,

J(U 0 ) ≤ δ n J(U -n ) + K ≤ K δ n (1 + 1 (∆x) 2 ) + 1 . ( 28
)
Let n → +∞ concludes the proof of the Proposition, using once again (24).

Remark 1. The proof of this proposition shows that there exists an absorbing ball in H 1 ∆x that captures all the trajectories in L 2 ∆x , but with a discrete time that depends on ∆x.

We now complete the proof of Theorem 1 by the following statement Proposition 4. A ∆x is a bounded subset in H 2 ∆x , uniformly in ∆x.

Proof : Introduce Z n+1 = U n+1e -α∆t U n ∆t that plays the role of the time derivative of U n . We infer from (8) that

Z n+1 -e -α∆t Z n ∆t -i 2 A Z n+1 + e -α∆t Z n + i 4 (|U n+1 | 2 + |U n | 2 ) U n+1 + e -α∆t U n -i 4 (|U n-1 | 2 + |U n | 2 ) U n + e -α∆t U n-1 = 0. ( 29
)
Multiply this equation by Z n+1 + e -α∆t Z n in the Hilbert space L 2 ∆x to obtain

Z n+1 2 L 2 ∆x ≤ e -2α∆t Z n 2 L 2 ∆x + c(∆t) U n 2 L ∞ ∆x + U n-1 2 L ∞ ∆x Z n+1 L 2 ∆x + e -α∆t Z n L 2 ∆x U n-1 L 2 ∆x + U n L 2 ∆x + U n+1 L 2 ∆x . (30)
Since we know that A ∆x is bounded in H 1 ∆x , uniformly in ∆x, proceeding as in the proof of Proposition 1 (see ( 9)) we obtain that there exists K that depends on α, F L 2 ∆x such that

Z n+1 2 L 2 ∆x ≤ δ Z n 2 L 2 ∆x + K∆t, (31)
since

Lemma 4. ∀v ∈ C N , v L ∞ ∆x ≤ c v H 1 ∆x . Therefore U n+1 -e -α∆t U n ∆t remains in a bounded set of L 2 ∆x . Then U n+1 + e -α∆t U n remains bounded in H 2 ∆x , uniformly in ∆x, i.e. ∃K = K(α, F L 2 ∆x ) ; K ≥ U n+1 + e -α∆t U n H 2 ∆x ≥ U n+1 H 2 ∆x -e -α∆t U n H 2 ∆x . (32)
Then, for any n U n+1

H 2 ∆x ≤ e -α∆t U n H 2 ∆x + K, (33)
and then U 0

H 2 ∆x ≤ e -nα∆t U -n H 2 ∆x + K ≤ e -nα∆t 1 + 4 (∆x) 2 U -n L 2 ∆x + K. (34)
If U n is a complete trajectory that is bounded in L 2 ∆x , n → +∞ completes the proof of the Theorem.

Remark 2. This result is sharp. The assumption F L 2 ∆x < +∞ uniformly in (∆x) implies that the stationary solutions belongs only to H 2 ∆x .

3. Consistency and convergence of the attractors.

3.1.

Existence and regularity of the attractor for the ODE. To begin with, we recall some results for the continuous attractor that were established in [START_REF] Goubet | Approximate inertial manifolds for the weakly damped nonlinear Scrödinger equations[END_REF].

Theorem 2. Consider a trajectory u(t) that is included in the continuous attractor A.

Then there exists a constant K that depends only on α, ||f || L 2 (T) such that for any

t in R ||u(t)|| H 2 (T) + ||u t (t)|| H 4 (T) + ||u tt (t)|| H 4 (T) ≤ K. ( 35
)
At a first glance, this result is surprising and means that along the invariant set the time derivative of the solution is more regular than the solution itself. Actually, one can go further in the regularity in t of the solutions of a PDE along an invariant set (see [START_REF] Hale | Regularity, determining modes and Galerkin methods[END_REF]). On the other hand, the estimate (35) provides upper bound for the distance between approximate inertial manifolds and A (see [START_REF] Goubet | Approximate inertial manifolds for the weakly damped nonlinear Scrödinger equations[END_REF] and the references therein).

We now go back to our numerical scheme. Consider (5) as a dynamical system in L 2 ∆x . One can prove the existence of a global attractor A ODE as in [START_REF] Yan | Attractors and dimensions for discretizations of a weakly damped Schrödinger equations and a sine-Gordon equation[END_REF] for instance. Moreover, in the spirit of Theorem 2, one can prove Theorem 3. Consider a trajectory U (t) that is included in the attractor A ODE . Then there exists a constant K that depends only on α, ||F || L 2 ∆x such that for any

t in R ||AU (t)|| L 2 ∆x + ||A 2 U t (t)|| L 2 ∆x + ||A 2 U tt (t)|| L 2 ∆x ≤ K. ( 36
)
Since the proof of the this theorem is just a suitable adaptation of the article [8] using discrete norms as in previous section, we omit it for the sake of conciseness.

3.2. Consistency. Consider U (t) a complete trajectory that is embedded in the attractor A ODE . Set V n U (n∆t). We define the consistency error as

η n = V n+1 -e -α∆t V n ∆t -iA V n+1 + e -α∆t V n 2 + i 4 |V n+1 | 2 + |V n | 2 (V n+1 + e -α∆t V n ) - 1 + e -α∆t 2 F. (37)
We now state and prove Proposition 5. There exists K that depends only on the data α, ||F || L 2 ∆x and that does not depend on ∆x such that ||η n || L 2 ∆x ≤ K(∆t) 2 . Proof: Integrate (5) between n∆t and (n+1)∆t to obtain (recalling that δ = e -α∆t )

V n+1 -δV n = (n+1)∆t n∆t e s-α(n+1)∆t F + iAU (s) -iU 2 (s)U (s) ds. (38)
We now invoke the well-known trapezoid formula to estimate the r.h.s of (38) We thus obtain

(n+1)∆t n∆t e αs-α(n+1)∆t F + iAU (s) -iU 2 (s)U (s) ds = ∆t 2 F + iAV n+1 -i|V n+1 | 2 V n+1 + δ(F + iAV n -i|V n | 2 V n ) + ζ n , (39)
where

||ζ n || L 2 ∆x ≤ c(∆t) 3 e -α(n+1)∆t sup t∈[n∆t,(n+1)∆t] ||∂ 2 t e αt (-iF + AU -|U | 2 U )(t) || L 2 ∆x .
Therefore, using Theorem 3, it is an exercise to prove that

e -α(n+1)∆t sup t∈[n∆t,(n+1)∆t] ||∂ 2 t e αt (-iF + AU -|U | 2 U )(t) || L 2 ∆x ≤ K .
To complete the consistency estimate, we have to majorize in L 2 ∆x the difference

(|V n+1 | 2 V n+1 +δ|V n | 2 V n )-( |V n+1 | 2 + |V n | 2 2 )(V n+1 +δV n ) = 1 2 (|V n+1 | 2 -|V n | 2 )(V n+1 -δV n ).
(40) Since by Theorem 3the time derivative U t is uniformly bounded, we bound this term by K(∆t) 2 . This completes the proof of the proposition.

3.3.

Upper semi-continuity of the attractors. Introduce I the interpolation operator that maps the continuous space into the discrete one by (Iu) j = 1 ∆x (j+1/2)∆x (j-1/2)∆x u(y)dy. We rather use this "L 2 " interpolation operator instead of the classical one since we only assume that f belongs to L 2 (T) . Anyway (Iu) j ∼ u(j∆x) if u is regular enough. The aim of this section is to establish that when both ∆t and ∆x converges to 0, then the Hausdorff semi-distance sup a∈A inf b∈A ∆x ||Ia -b|| L 2 ∆x converges to 0. The idea is to use A ODE and the following triangle inequality

sup a∈A inf b∈A ∆x ||Ia -b|| L 2 ∆x ≤ sup c∈A ODE inf b∈A ∆x ||c -b|| L 2 ∆x + sup a∈A inf c∈A ODE ||c -Ia|| L 2 ∆x . ( 41
)
Consider the first term in the r.h.s of (41). Consider U (t) a complete trajectory in

A ODE . Set G(U n , U n+1 ) = F -i 4 (|U n | 2 + |U n+1 | 2 )(δU n + U n+1 ). Then, if V n = U (n∆t) and ε n = U n -V n , ε n+1 -δε n ∆t + i 2 A(ε n+1 + δε n ) = G(U n , U n+1 ) -G(V n , V n+1 ) + η n , (42)
where η n is the consistency error. Using the upper bounds on the attractors given in Theorems 1 and 3, one can prove that

||G(U n , U n+1 ) -G(V n , V n+1 )|| L 2 ∆x ≤ K||ε n+1 + δε n || L 2 ∆x . Actually U n is not in the discrete attractor, but since U 0 = V 0 is in H 2 ∆x
and since the equation features also an H 2 ∆x absorbing set, we may assume that U n is bounded in H 2 ∆x . Considering then the scalar product of 42 with ε n+1 + δε n in L 2 ∆x we thus obtain

||ε n+1 || 2 L 2 ∆x -δ 2 ||ε n || 2 L 2 ∆x ≤ K∆t ||ε n+1 + δε n || 2 L 2 ∆x + ||η n || 2 L 2 ∆x , (43)
and then, by the consistency estimate of Lemma 5,

||ε n+1 || L 2 ∆x ≤ δ 2 1 + K∆t 1 -K∆t ||ε n || L 2 ∆x + K(∆t) 3 . ( 44
)
Using δ 2 1+K∆t 1-K∆t ≤ e 2(K-α)∆t = β for ∆t small enough, we then have by the discrete Gronwall lemma, (and with ε 0 = 0)

||ε n || L 2 ∆x ≤ β n K (∆t) 3 β -1 ≤ Kβ n (∆t) 2 . ( 45
)
We now infer from (45) Proposition 6.

lim ∆t→0 sup c∈A ODE inf b∈A ∆x ||c -b|| L 2 ∆x = 0. (46)
Proof: Consider U (t) a complete trajectory in A ODE as above. Consider estimate (45) between U (0) and U n = S n (U (-n∆t)), where S stands for the discrete semi-group that is defined by U n+1 = SU n where U n+1 solves [START_REF] Goubet | Approximate inertial manifolds for the weakly damped nonlinear Scrödinger equations[END_REF]. Consider ε > 0. To the very definition of a global attractor we know that there exists b in A ∆x such that for n large enough

||b -S n (U (-n∆t))|| L 2 ∆x ≤ ε.
(47) Fix then n in (45) and let ∆t → 0 that gives ||b-U (0)|| L 2 ∆x ≤ ε. Since ε is arbitrary the proof of the Proposition is completed. It remains to compare the attractors A and A ODE . We give an overview of this process that is more or less standard. Consider a complete trajectory u(t) in the global attractor A. We apply the interpolation operator I to the equation ( 1). This gives,

(Iu) t + αIu + iI(u xx ) + iI(|u| 2 u) = If = F.
(48) Let us point out that the operator I and ∂ x do not commute; this shows in the consistency error as follows

η(t) = i(AIu -I(u xx )) + i(|Iu| 2 Iu -I(|u| 2 u));
(49) observe that due to standard results and since u(t) is bounded in . We conclude then as in the proof of Proposition 6.

H 2 (T), then ||η(t)|| L 2 ∆x = O(1) when ∆x → 0, uniformly in t. Introduce now V (t) = U (t) -Iu(t) that solves V t + αV -iAV = η + i(|Iu| 2 Iu -|U | 2 U ). (50)

Numerical experiments.

4.1. Implementation of the scheme. The implementation of the Crank-Nicolson scheme needs a fixed point problem to be solved at each time step. Let Id be the Identity matrix. If we set

Φ(U, U n ) = Id -i ∆t 2 A -1 {ǫ -α∆t (Id + i ∆t 2 A)U n -i∆t 4 (|U | 2 + |U n | 2 ) U + e -α∆t U n + 1 + e -α∆t 2 F }
then the time marching scheme reduces to solve the fixed point problem

U = Φ(U, U n ) (51)
at the n-th time step. Lemma 1 insures that there exists a unique fixed point if α∆t is small enough, the proof uses the classical Banach fixed point theorem. In practice, for this NLS equation, the convergence of the Picard iterates is obtained by taking only very small values of ∆t, typically ∆t ≃ 10 -4 . This is dramatic since we are looking to the long time numerical behavior of the solution. Anyway, this effective restriction on the time step is really artificial because the scheme is supposed to be unconditionally stable. For this reason, we propose to solve (51) by accelerating the (Picard) sequence

U (0) = U n for m = 0, • • • U (m+1) = Φ(U (m) , U n ), (52)
enhancing in that way the stability region, allowing then to take larger values of ∆t. To this end, we use the ∆ k acceleration procedure, see [START_REF] Brezinski | Nonlinear hybrid procedures and fixed point iterations[END_REF] and the references therein. In two words, the ∆ k procedure consists in replacing the Picard iterates by

U (0) = U n for m = 0, • • • U (m+1) = U (m) -(-1) k α k m ∆ k Φ U (m) ; (53)
where

∆ k Φ U (m) = k j=0 C k j (-1) k-j Φ (j) (U (m) , U n ), C k j = k! j!(k -j)!
is the binomial coefficient and Φ (j) denotes the j-th composition of Φ with itself. The parameter α k m is computed such as minimizing the euclidian norm of the linearized part of the residual r (m+1) = u (m+1) -Φ(U (m) , U n ). We have

α k m = (-1) k < ∆ 1 φ U (m) , ∆ k+1 Φ U (m) > < ∆ k+1 Φ U (m) , ∆ k+1 Φ U (m) > (54)
see [START_REF] Brezinski | Nonlinear hybrid procedures and fixed point iterations[END_REF]. These acceleration procedure have been applied with success for solving nonlinear eigenvalue problem and presently, for the NLS scheme, we obtain good results with the ∆ 1 (Lemaréchal's) and the ∆ 2 (Marder-Weitzner) methods; we can take ∆t up to 1.10 -1 and after a transient time very few iterations are needed for solving (51) with a good accuracy. To achieve the description of the implementation of the scheme (8), we mention that the evaluation of Φ necessitates the solution of a linear system with the matrix Idi ∆t 2 A. Since this matrix is independant of t, this is done by a QR factorization method. 4.2. Discrete smoothing. We address here the following issue : consider U ∈ C N , that could be the sampling of a function at points j∆x. How to check that this sampling corresponds to a function that belongs to H 1 ? In a perfect world, we just have to check that the sequence :

α N = 1 ∆x N j=1 (u j+1 -u j ) 2 (55)
remains bounded when N → +∞. In the real world of numerics, we can deal with large N's but hardly pretend that ∆x = 1 N → 0. Therefore we shall use in the sequel the following ansatz : consider a periodic function to compute the coarser grid approximation, we make use of the formula

σ(x) = σ(2x) + 1 2 (σ(2x + 1) + σ(2x -1)). ( 60
)
Therefore U expands / splits as follows

u(x) = m j=1 U 2j σ x ∆x -2j + m j=1 U 2j+1 σ x ∆x -2j -1 = m j=1 U 2j σ x 2∆x -j - 1 2 m j=1 U 2j σ x ∆x -2j + 1 + σ x ∆x -2j -1 + m j=1 U 2j+1 σ x ∆x -2j -1 = m j=1 U 2j σ x 2∆x
j approximation on the coarse grid

+ m j=1 U 2j+1 - U 2j + U 2j+2 2 σ x ∆x -2j -1 difference (61) This corresponds to --X --- ---X --- ↓ 1 -1 2 ց ↓ 1 ւ -1 2 ---X --- ---X ---
We now define the incremental unknown as

z j = U 2j+1 - U 2j + U 2j+2 2 , (62)
We compute recursively the incremental unknowns corresponding to nested grids from a coarser grid to the finest one. The criterion reads

k=1 4 ks ||w k || 2 L 2 , < +∞ (63)
where w k is the incremental unknown corresponding to the grid of mesh size 2 -k . Actually, if we compute the convergence radius ρ of the series

||w k || 2 L 2 Z k in C, then ρ = 4 s .
4.4. Preliminary tests. Before to go further on the investigation of the discrete regularity of the solutions of our discrete nonlinear Schrödinger equations, we will check that both the Fourier and multigrid approach allow us to compute successfully the discrete regularity of the following test functions:

• f 1 (x) = -1 if x ∈ [0, 1 2 [ 1 if x ∈ [ 1 2 , 1[ • f 2 (x) = |x -1 2 | • f 3 (x) = x 2 (1 -x) 2 • f 4 (x) = sin(2πx).
We proceed to the computations on a grid with N=128 points (x i , y i ) for 1 ≤ i ≤ N such that x i = ih, ∆x = 1 N , and y i = f l (x i ) with the four functions f l ; l = 1, 2, 3, 4. defined above. We aim to compute the regularity index s l of each function f l , both with Fourier analysis and multigrid approaches. Then f l belongs to H s ∆x with Fourier Multigrid analysis approach f 1 s=0.6049 s=0.50 f 2 s=1.6357 s=1.50 f 3 s=3.7917 s=2.00 f 4 s=10.000 s=1.9834 These results show that we are able to give a satisfactory measurement of the regularity index for a given function, but with a limitation of our multigrid approach. Actually, since our incremental unknowns behaves as wavelet coefficients with a wavelet that has two vanishing moments, we cannot go further than 2 in the regularity measurement process. Moreover, we can explain the error on the measurement of the regularity of the analytical function f 4 in the Fourier analysis setting by round up errors in the numerical computations. 4.5. Numerical evidences for the regularity of solutions to discrete Schrödinger equation. We compute below the discrete regularity of a solution of (8) and we plot the evolution of this discrete regularity with respect to time. The computations are performed for different forcing terms f and for various initial data u 0 . In the computations below α = 0.01 and the time step ∆t is equal to 10 -2 .

The figures in the sequel represent the regularity of the solution as a function of time, both with Fourier and multigrid approaches. All the example below give evidence of robustness and stability of our Crank-Nicolson scheme for long-time computations.

• To begin with, we are given a smooth forcing term f = 0 and a smooth initial data u 0 = sin(2πx). We observe that in that case there is no loss of regularity along the process (see Fig 1). Once again, the measurement of regularity is limited when using multigrids.

• We now introduce a forcing term f that is the Heaviside function (regularity H • In that example the forcing term is a L 2 function obtained with a random procedure and the initial data is smooth. The regularity of the solution converges towards the expected value of the regularity of the attractor that is 2 in this case. • This example is similar with the previous one, but we are given here a non-smooth initial data u 0 that is the Heaviside function. Once again, the regularity of the solution converges towards the expected value of the regularity of the attractor that is 2 in this case.

Lemma 5 .

 5 Consider g(t) a smooth function. Then | b a g(s)ds -b-a 2 (g(b) + g(a))| ≤ c(ba) 3 sup t |g(t)|.

Figure 1 .

 1 Figure 1. Regularity of u(t, •) when α = 0.01, f = 0, u 0 (x) = sin(2πx).

Figure 2 .

 2 Figure 2. Regularity of u(t, •) when α = 0.01, f = Heaviside, u 0 ≡ 1.data we perform a random function u 0 . The results are very interesting. After a transient time the solution forget the low regular initial data and its regularity converges to the expected regularity of the attractor as in the example above.

  , f=heaviside, u 0 ∈L 2 Fourier Multigrid

Figure 3 .

 3 Figure 3. Regularity of u(t, •) when α = 0.01, f Heaviside, u 0 ∈ L 2 .

Figure 4 .

 4 Figure 4. Regularity of u(t, •) when α = 0.01, f ∈ L 2 , u 0 ≡ 1.

Figure 5 .

 5 Figure 5. Regularity of u(t, •) when α = 0.01, f ∈ L 2 , u 0 Heaviside.

  Considering the scalar product in L 2 ∆x with w, we thus obtain, since |||Iu| 2 Iu -|U | 2 U || L 2 ∆x ≤ K||V || L 2 ∆x in the global attractor that ||V (t)|| L 2 ∆x ≤ e Kt sup t ||η(t)|| 2

	L 2 ∆x

Acknowledgements. This work initiated when the two last authors were visiting the University Cadi Ayyad of Marrakech and has been partially supported by the research programs "Dissipation and asymptotique" and "Couplage Fluide-Sable" by CNRS (France) and CNRST (Marocco), and also by the INRIA research program MASOH (Modélisation, Analyse et Simulation d'Ondes Hydrodynamiques).

that expands as a Fourier series

where e 2k (x) = sin(2kπx) , k ≥ 1 , e 2k+1 (x) = cos(2kπx) , k ≥ 0. The smoothness of the function u(x) shows on the decay of the high frequency modes.

In the numerical computations we shall use the following criteria, that is standard in multilevel signal analysis (see [START_REF] Mallat | A wavelet tour of signal processing[END_REF]). A function u that is in L 2 belongs to H s , s > 0 iff

where

The last issue to address is to implement this criterion on U ∈ C N , pretending that U = (U j ) is the sampling of a periodic function at points j∆x. For that purpose we use the Discrete Fourier Transform

U j e -ikj∆x .

is the first approximation of U N , and the other ones are computed recursively. 4.3. Discrete smoothing : a multigrid approach.

Consider a fine grid with N = 2m points. Consider a sampling U = (U j ) 1≤j≤N ∈ C N . Let us pretend that U is represented by

where σ(x) = max(0, 1 -|x|) is the classical hat function. Proceeding as above, one may pretend that u is smooth if u is accurately represented by its approximation on the coarser grid with m points defined as follows :

---X ------X ------X ------X ---0 coarser grid point 1