Pharm.D Christophe Piérard
email: cpierard@imassa.fr

CA Pierrette Liscia

Jean-Nicolas Philippin

Nicole Mons

Thierry Lafon

Frédéric Chauveau

Pascal Van Beers

Isabelle Drouet

André Serra

Jean-Claude Jouanin

Daniel Béracochéa

Modafinil restores memory performance and neural activity impaired

Keywords: Modafinil, sleep deprivation, memory and neural activity modafinil, sleep deprivation, memory, neural activity

Introduction:

Modafinil (chemical name: [(diphenylmethyl) sulfinyl]-2 acetamide) has been reported as having stimulant and awakening properties without amphetamine-like side effects [START_REF] Bastuji | Successful treatment of idiopathic hypersomnia and narcolepsy with modafinil[END_REF][START_REF] Hermant | Awakening properties of modafinil : effect of nocturnal activity in monkeys (Macaca mulatta) after acute and repeated administration[END_REF][START_REF] Lyons | Modafinil : the unique properties of a new stimulant[END_REF]. Modafinil (Modiodal®) is successfully used in the treatment of narcolepsy and idiopathic hypersomnia without interfering, however, with nocturnal sleep [START_REF] Bastuji | Successful treatment of idiopathic hypersomnia and narcolepsy with modafinil[END_REF]. When administered to healthy subjects, modafinil provides a military interest in cases of total or partial sleep deprivation resulting from either continuous or sustained operation (Lagarde et al, 1995 ;Lagarde and Batejat, 1995 ;[START_REF] Caldwell | Modafinil's effects on simulator performance and mood in pilots during 37 hours without sleep[END_REF].

Its mechanism of action could involve both norepinephrine and dopaminergic systems, through ascending pathways that are likely to promote wakefulness by activating the cortex and other forebrain targets, possibly through interaction with the hypocretin/orexin system [START_REF] Boutrel | What keeps us awake : the neuropharmacology of stimulants and wakefulness-promoting medications[END_REF][START_REF] Willie | Modafinil more effectively induces wakefulness in orexin-null mice than in wild-type littermates[END_REF]. Moreover, modafinil is able to interact with norepinephrine and dopamine transporters in the brain [START_REF] Madras | Modafinil occupies dopamine and norepinephrine transporters in vivo and modulates the transporters and trace amine activity in vitro[END_REF], and to enhance the extracellular level of dopamine in the nucleus accumbens in rats [START_REF] Murillo-Rodriguez | Modafinil enhances extracellular levels of dopamine in the nucleus accumbens and increases wakefulness in rats[END_REF], probably via the involvement of a GABAergic mechanism [START_REF] Ferraro | The vigilance promoting drug modafinil increases dopamine release in the rat nucleus accumbens via the involvement of a local GABAergic mechanism[END_REF]. Moreover, behavioural activity induced by modafinil administration could be mediated by brain alpha 1B-adrenoceptors [START_REF] Stone | Role of brain alpha 1B-adrenoceptors in modafinil-induced behavioural activity[END_REF]. The mechanism of action of modafinil could also involve a reduction of GABA release in the cerebral cortex [START_REF] Tanganelli | Inhibitory effects of the psychoactive drug modafinil on gamma-aminobutyric acid outflow from the cerebral cortex of the awake freely moving guinea-pig. Possible involvement of 5-hydroxytryptamine mechanisms[END_REF][START_REF] Piérard | Effects of a vigilance-enhancing drug, modafinil, on rat brain cortex amino acids : a microdialysis study[END_REF] as well as an involvement of excitatory amino acids system [START_REF] Piérard | Effects of a vigilance-enhancing drug, modafinil, on rat brain metabolism : a 2D 1 H-NMR study[END_REF]1997) and its receptors [START_REF] Lagarde | Modulation of the stimulating effect of modafinil by glutamate agonists and antagonists[END_REF].

Modafinil could also enhance brain energy metabolism, by increasing the energetic pool of phosphocreatine in cortex, thus contributing vigilance-enhancing properties of this drug [START_REF] Piérard | Effects of a vigilance-enhancing drug, modafinil, on rat brain metabolism : a 2D 1 H-NMR study[END_REF]. Moreover, we recently demonstrated that modafinil administered in chronically-stressed mice could interfere with glucocorticoids [START_REF] Piérard | Modafinil-induced modulation of working memory and plasma corticosterone in chronically-stressed mice[END_REF], that are involved in the regulation of dopaminergic neurotransmission [START_REF] Czyrak | Role of glucocorticoids in the regulation of dopaminergic neurotransmission[END_REF]. However, the mechanism of action of modafinil,--as yet not fully understood--, remains largely surrounded by controversy [START_REF] Saper | Modafinil : a drug in search of a mechanism[END_REF].

Concerning the neuroanatomical brain targets of modafinil, some authors observed that c-Fos protein expression increased in the anterior hypothalamus following modafinil administration [START_REF] Lin | Potential brain neuronal targets for amphetamine, methylphenidate, and modafinil-induced wakefulness, evidenced by c-fos immunocytochemistry in the cat[END_REF][START_REF] Engber | Differential patterns of regional c-Fos induction in the rat brain by amphetamine and the novel wakefulness-promoting agent modafinil[END_REF] and orexin-containing neurons [START_REF] Scammel | Hypothalamic arousal regions are activated during modafinil-induced wakefulness[END_REF]. Neither one of these studies, however, was performed after a sleep deprivation period followed by cognitive tasks; hence the brain targets of modafinil remain a matter for debate [START_REF] Gallopin | Effect of the wake-promoting agent modafinil on sleep-promoting neurons from the ventrolateral preoptic nucleus : an in vivo pharmacologic study[END_REF].

From a functional viewpoint, specific neurocognitive domains including executive attention, working memory and higher cognitive functions are particularly vulnerable to sleep loss [START_REF] Durmer | Neurocognitive consequences of sleep deprivation[END_REF][START_REF] Mu | Decreased brain activity during a working memory task at rested baseline is associated with vulnerability to sleep deprivation[END_REF]. More specifically, sleep loss, even moderate, compromises the function of neuronal circuits critical to subsecond attention allocation during working-memory tasks [START_REF] Smith | The impact of moderate sleep loss on neurophysiologic signals during working-memory task performance[END_REF]. The associated decrease in brain activity during working-memory tasks appears to be a function of the individual sleep-deprivation vulnerability [START_REF] Mu | Decreased brain activity during a working memory task at rested baseline is associated with vulnerability to sleep deprivation[END_REF].

Moreover, executive functioning is largely dependent on activity in the prefrontal cortex and affected negatively by one night sleep deprivation [START_REF] Nilsson | Less effective executive functioning after one night's sleep deprivation[END_REF].

In addition, several studies without sleep deprivation have demonstrated that modafinil is able to improve working memory both in animals [START_REF] Béracochéa | First evidence of a delay-dependant working memory-enhancing effect of modafinil in mice[END_REF] as well as in humans [START_REF] Müller | Effects of modafinil on working memory processes in humans[END_REF]. There is further evidence that modafinil enhances learning processes after chronic [START_REF] Béracochéa | Improvement of learning processes following chronic systemic administration of modafinil in mice[END_REF] or acute [START_REF] Béracochéa | Enhancement of learning processes following an acute modafinil injection in mice[END_REF] systemic administration in mice. To the best of our knowledge, very few studies investigated the effect of modafinil administration on memory and/or neural activity after sleep deprivation. In one of these studies [START_REF] Pigeau | Modafinil, damphetamine and placebo during 64 hours of sustained mental work. I. Effects on mood, fatigue, cognitive performance and body temperature[END_REF], however, modafinil seems to restore short-term memory after prolonged sleep deprivation in humans.

Hence, the original aims of our study have been to investigate in sleep-deprived mice, the effects of modafinil on spatial working memory with subsequent evaluation of the neural activity level in various cerebral zones, as compared to non sleep-deprived animals.

Materials and Methods:

Animals:

The study was conducted using male mice of the C57BL/6J strain (Iffa-Credo, Lyon, France).

Upon arrival, mice were housed collectively in colony cages (40 cm long x 25 cm high x 20 cm wide) matched for weight and placed in an animal room (22°C ambient temperature ; automatic light cycle 07:00 a.m. and 07:00 p.m.) with free access to food and water. Fifteen days before testing, mice were placed in individual cages and manipulated 10 minutes each day, in order to reduce interference with the experimenter. They were 5 months old on the day of the experiment.

The present study was carried out in compliance with the European Convention for the protection of Vertebrate Animals used for Experimental and other Scientific Purposes, and under the agreement # 94001 delivered by the French Ministry of Defense, after the protocol was examined by the local ethical committee.

Sleep deprivation model:

Apparatus:

An original automatic total sleep-deprivation apparatus ("water box") was developed jointly by LNC and IMASSA in order to reduce stress and physical fatigue [Fig. 1]. It is constituted by a water tank of grey PVC (10 mm thickness) with rectangular section (42 x 32 cm) and 22 cm high, containing temperature-regulated water at 31° C. In the middle of the tank two small square platforms (10 x 10 cm), without edge, are closely adjusted side by side. The surface of each platform is squared with deep (3 mm) orthogonal grooves in order to facilitate water evacuation during upward movements, and consequently to avoid that animals slip or get their feet wet. The platforms move independently with compressed-air jack. In the start position, both platforms slightly emerge from water level. Then, each platform alternatively moves below and above the surface of the water, thus forcing the mouse to a permanent motion in order to avoid water contact. The completion of one platform revolution needs 10 sec. During this period, each platform remains stable over the surface of the water for 5 sec, then goes 60 mm down for 2.5 sec and immediately goes up for 2.5 sec until the initial position. Throughout the days prior to the behavioural task, the mice were allowed to become familiar with the water box to reduce their stress level. Thus, they learn to stay at the junction of both platforms, allowing them to move from the sinking platform with a very short motion (i.e. 1 step forward then backward every 10-sec period).

Validation of sleep deprivation model by EEG recording:

The determination of the efficacy of our original sleep deprivation apparatus was performed on 14 independent mice, divided into a 10-hr diurnal sleep-deprived group (n = 7) and a control (non sleep-deprived) group (n = 7). All the mice were monitored by means of continuous video and EEG recording for 10 hours, between 08:00 a.m. and 08:00 p.m. Control mice were also placed in the sleep deprivation apparatus during the same period, with both platforms unmoved, thus allowing them to sleep ad libitum. Eight days before the validation experiment, animals were anaesthetized in order to place in parietal position 2 silver wire electrodes close to the dura-matter. Moreover, one reference electrode was inserted in the frontal bone. All the electrodes were fixed using acrylic dental cement (Palavit®). The animals were then allowed to recover for 8 days. On the day of the experiment, the animals were placed in the sleep deprivation apparatus. Cortical electric activity was recorded on a Medelec® 1121 EEG table. Further, spectral analysis was performed with the Somnologica® v.3.1.3 software. In addition to the observation of animals, the wakening status was characterized on recordings by low amplitude (< 200 µV) and high frequency (> 40 Hz) waves, and the sleep status by the occurrence of waves with an amplitude higher than 200 µV, and of δ (0.5 -4.5 Hz) or θ (4.5 -9.5 Hz) frequencies. Signal analysis was performed continuously throughout the 10 hrs of sleep deprivation, for 5 s periods. Moreover, selective spectral analysis was performed for power in the (8-9 Hz) θ band frequencies, considered as an index of alertness or attention during wakefulness in the rodents [START_REF] Willie | Modafinil more effectively induces wakefulness in orexin-null mice than in wild-type littermates[END_REF].

Memory testing:

The behavioural task used to test working memory [Fig. 2] is based on spontaneous alternation behavior (SA), which does not require the use of food reinforcement to emerge. SA is the innate tendency of rodents whereby over a series of trials run in a T-maze, they alternate at each successive trial the choice of the goal arm ((except for the first trial). Repetitive testing constitutes a potent source of proactive interference. From trial to trial, accurate performance at a given trial (N) requires for subjects to be able to discriminate the specific target trial N-1 from the interfering trial N-2. The target information required for successful performance varies from trial to trial, so that the subject is not only required to keep temporarily in short-term memory specific information, but also to reset it over successive runs. The resetting mechanisms and cognitive flexibility required to alternate over successive runs are major components of working memory processes. Working memory is a component of the sequential alternation task, since SA rates are dependent on the length of the intertrial delay interval, and/or the place of the trial in the series. Indeed, repetitive testing constitutes a potent source of proactive interference. Thus, the sequential alternation procedure is relevant to assess delay-dependent working memory in mice [START_REF] Béracochéa | Effects of ibotenic acid lesions of the mammillary bodies on spontaneous and rewarded spatial alternation in mice[END_REF][START_REF] Béracochéa | Methyl beta-carboline-3-carboxylate reverses the working memory deficits induced either by chronic alcohol consumption or mammillary bodies lesions in mice[END_REF].

The tests were carried out in a T-maze constructed of opaque grey PVC. Stem and arms were 35 cm long, 10 cm wide, and 25 cm high. The start box (10 x 12 cm) was separated from the stem by a vertical sliding door. Vertical sliding doors were also placed at the entrance of each arm. A low-intensity diffuse illumination (10 lux) was provided above the apparatus. Between two trials, the apparatus was cleaned using 70 % alcohol and water, in order to remove any olfactive cue. An alternation response was considered each time the subject entered the arm opposite to the one visited on the immediately previous trial. Alternation rate was expressed in percentage relative to the maximal alternation rate of 100 % (obtained when the subject never returned into the same arm over two consecutive trials).

For all the animals, the following procedure was used:

On D-4 (four days before the day of the test), all the animals were allowed to freely explore the apparatus 10 min in the morning and 10 min in the afternoon, in order for them to become familiar with the experimental conditions. Between both sessions, animals were replaced in their individual cage in the animal room.

From D-3 to D-1, the animals had to complete two daily training sessions of SA (6 sessions in total), in order to foster the development of the alternation behavioural pattern and to familiarize them with the opening and/or closing of the doors over successive runs. Each session included 6 trials, separated by a 30-s intertrial delay.

On D0 (the day of the experiment), behavioural testing was performed 30 min after the end of the sleep deprivation period and modafinil (or vehicle) injection. This delay is required for the development of the pharmacological action of modafinil. The animals were not allowed to sleep during this 30-min delay, because recovery sleep might interfere with Fos staining [START_REF] Cirelli | How sleep deprivation affects gene expression in the brain : a review of recent findings[END_REF]. As stated here above, all the subjects were exposed to 6 successive trials (intertrial delay: 30 s).

Behavioural study:

Previous experiments have shown that a sleep deprivation period of 24 hrs totally disrupted the ability of mice to run the alternation task, the mice choosing to sleep rather than to explore the maze. In contrast, a 3-hr sleep deprivation period had no effect on memory performance as regards non sleep-deprived animals (unpublished data). According to these previous experiments, we decided to study the effects on memory of an intermediate 10-hr sleep deprivation period and modafinil. Thirty-eight mice randomized between the 5 following groups were used: i) NSD (non sleep-deprived ; n = 8) ; ii) SD (10-hr sleep-deprived ; n = 8) ; iii) SD + vehicle (10-hr sleepdeprived + vehicle ; n = 6) ; iv) SD + M32 (10-hr sleep-deprived + modafinil 32 mg/kg ; n = 8) ; v) SD + M64 (10-hr sleep-deprived + modafinil 64 mg/kg ; n = 8). The sleep deprivation period took place from 08:00 a.m until 08:00 p.m.

Modafinil administration: modafinil, insoluble in water, was suspended in an 0.5 % solution of gum tragacanth (vehicle). Modafinil suspension or vehicle alone were injected i.p. after the sleep deprivation period, 30 minutes before the behavioural test session began. According to our previous study [START_REF] Béracochéa | First evidence of a delay-dependant working memory-enhancing effect of modafinil in mice[END_REF][START_REF] Piérard | Modafinil-induced modulation of working memory and plasma corticosterone in chronically-stressed mice[END_REF], the administered doses of modafinil were of 32 and 64 mg/kg for the M32 and M64 groups respectively (injected volume: 0.1 ml/10 g mouse).

Statistical analysis: in our behavioural procedure, only mice having alternated at the second trial were selected for further statistical analysis of behavioural data [START_REF] Piérard | Modafinil-induced modulation of working memory and plasma corticosterone in chronically-stressed mice[END_REF]. This behavioural criterion ensured that motor abilities and motivation to alternate are equivalent between subjects and not impaired by sleep deprivation and/or modafinil. Thus, all the groups exhibit alternation rates of 100 % at the second trial [Fig. 5]. Using this procedure, only one mouse of the [SD] group and one mouse of the [SD + M32] group were not taken into account for analysis.

Moreover, mice of the [SD] and [SD + vehicle] groups were gathered together for statistical analysis. Statistical analysis was performed using Statview® v.5.0 software. ANOVA was performed to evidence the effects of a 10-hr sleep deprivation period (followed or not by modafinil administration) on alternation rates. Further comparisons between individual groups were performed with the Scheffe post-hoc test. Significance was accepted at p values below 0.05.

Immunohistochemical study (c-Fos analysis):

The quantification of c-Fos protein expression in the brain is currently used as a marker of functional activity (see Krukoff 1999 for review). C-Fos analysis was performed on the 10-hr sleep-deprived groups, as compared to the control non sleep-deprived group [NSD]. Immediately after the alternation task, 12 mice of [NSD], [SD], and [SD + M64] groups (4 randomized mice for each group) were anaesthetized (ketamine 100 mg/kg + xylazine 10 mg/kg ; injected volume: 0.1 ml/10 g mouse). They were then perfused in the left ventricule with 100 ml NaCl 0.9 % followed by 100 ml paraformaldehyde 4 % to fix tissue before the brain was removed. Fixed brains were cut in frontal sections of 50 µm each, using a freezing microtome. Sections were incubated with a primary antibody specific of Fos protein (Calbiochem®), then with a biotinyled secondary antibody (specific of the primary antibody), and finally with the avidine-biotine-peroxydase complex Vectastain®). C-Fos immunoreactivity was revealed using diaminobenzidine. Different brain structures were analysed with the image analysis system Biocom 2000 (visiol@b 2000 software) coupled with a microscope. The analysis concerned the brain structures involved i) in the regulation of the wake-sleep cycle : anterior hypothalamus and supraoptic nucleus, ii) in memory : frontal cortex and hippocampus areas (CA1 and dentate gyrus) and iii) in emotions : amygdala.

The quantification of Fos-positive nuclei in the studied brain regions following the alternation task was expressed in mean counts/mm 2 . Moreover, for a given area, the results were expressed in relative variations as compared to [NSD] mice. Intra-zone comparisons to [NSD] mice were performed by the Student t test. Significance was accepted at p values below 0.05.

Results:

EEG validation of the sleep deprivation model:

The behavioural observation of video recordings obtained during the 10-hr sleep deprivation period shows that mice remained awake 100 % of the time in the water-box. The [Fig. 3] represents an EEG sample obtained during the occurrence of a short sleep event. The [Fig. 4], obtained from EEG data, gives the repartition (in terms of numbers and mean durations of sleep events) during the 10-hr diurnal sleep deprivation period. The mean cumulated duration of sleep episodes during the 10-hr sleep deprivation period was of about 0.66 ± 0.23 min (m ± SEM ; n = 7), i.e. 0.11 ± 0.04 %, whereas it was of 314.13 ± 27.56 min (m ± SEM ; n = 7), i.e. 52.36 ± 4.59 % in control animals.

Thus, the efficacy of our sleep-deprivation apparatus is of 99.79 %. The greatest number and duration of sleep episodes occurred during the first hour spent on the apparatus (between 08:00 and 09:00 a.m.), i.e. at the beginning of the diurnal rest phase for rodents. On the other hand, spectral analysis of the θ band frequencies (8-9 Hz) during wakefulness shows that attention is continuously impaired from the first hour of the sleep deprivation (08:00-09:00 a.m.) until the fourth hour (11:00-12:00 a.m.). Minimal attention also occurred for the seventh hour (02:00-03:00 p.m.). Then attention increased until the end of the sleep deprivation period (i.e. 05:00-06:00 p.m.). Furthermore, individual comparisons show that the alternation rate of the [SD] group is significantly decreased (p < 0.001) as compared to the [NSD] group, thus evidencing the impairment spatial working memory induced by a 10-hr total sleep deprivation period.

Effects of sleep deprivation:

On neural activity: [Table I]

For a given area, ANOVA performed on groups (SD, NSD, SD+M64) showed a global significant inter-groups difference as mentioned in Table 1 (p<0.05 in all comparisons). The mice of the [SD] group show a significantly decreased c-Fos staining in the anterior hypothalamus and supraoptic nucleus (-23 and -18 % respectively ; p < 0.05), as compared to [NSD] mice.

Moreover, the decrease in c-Fos staining is also evidenced in CA1 and dentate gyrus of hippocampus (-18 % ; p < 0.05), frontal cortex (-15 % ; p < 0.03) and amygdala (-13 % ; p < 0.05), as compared to [NSD] mice. group is significantly increased as compared to the [SD] group (p < 0.001), but not significantly different from the [NSD] group. Thus, modafinil (64 mg/kg, but not 32 mg/kg) is able to restore the spatial working memory performance that was previously impaired by a 10-hr sleep-deprivation period.

Effects of modafinil after

On neural activity: [Table I]

For all the studied brain regions, statistical analysis shows that c-Fos staining for the [SD + M64] group is not different as compared to [NSD] group. Thus, modafinil (64 mg/kg) is able to restore neural activity in the very same cerebral structures where it was previously decreased by a 10-hr sleep deprivation period.

Discussion:

The current study evidenced that a total 10-hr diurnal sleep deprivation is able to impair working memory performance in mice. Moreover, the 10-hr sleep deprivation period is able to impair neural activity in the cerebral areas involved in sleep/wake cycle regulation, memory and emotions. Both these impairments could be completely reversed by modafinil administration at the dose of 64 mg/kg i.p. (but not of 32 mg/kg).

Thus, the present study shows for the first time that modafinil is able to restore both memory performance and neural activity previously impaired by a prolonged sleep deprivation. Besides pharmacological means (for example IMAO that completely suppresses paradoxical sleep), or gentle handling [START_REF] Palchykova | Sleep deprivation impairs object recognition in mice[END_REF], several sleep deprivation apparatuses (mainly for rats or bigger animals) are described in the literature [START_REF] Vogel | A review of REM sleep deprivation[END_REF][START_REF] Rechtschaffen | Sleep deprivation in the rat : X. Integration and discussion of the findings[END_REF]. Furthermore, some of them allow selective REM or non REM sleep deprivation, with the help of polygraphic recordings. In the present study, we used an original total sleep deprivation model. We can assume that stress in our sleep deprivation model is not a cause for the memory impairments observed in SD subjects. Indeed, the advantage of our alternative platforms apparatus (water box) is to induce very low stress level. Thus, at any moment the animal has an emerging security platform readily available, allowing it to stay out of water. Furthermore, physical fatigue is minimal because the animal learns to stay at the junction of both platforms, thus allowing it to leave the downward platform very quickly, with a very short motion. In addition, the stress effect is observable only during the first hour of the SD period, and we found that mice submitted to a 3 hours SD period did not evidenced memory impairments (unpublished data). This means that the stress component involved in the first hour is not important enough to induce a memory deficit. Continuous EEG analysis (performed on independent mice during the validation phase) allowed to determine the efficacy score (99.79 %) of our sleep deprivation apparatus. On the other hand, EEG recording allowed to evidence variations of attention by the means of the spectral analysis of the θ band frequencies (8-9 Hz) during wakefulness.

In humans, modafinil administered in non sleep-deprived subjects at a therapeutic dose (100 or 200 mg) has no effect on spatial working memory, nor on long-term memory. Thus, modafinil could not be considered as a cognitive enhancer in non sleep-deprived healthy subjects [START_REF] Randall | Does modafinil enhance cognitive performance in young volunteers who are not sleep-deprived ?[END_REF]. On the opposite, previous results of our laboratory obtained on healthy sleepdeprived adults (after a 60-hr sleep deprivation period) showed that modafinil is able to restore the overall psychomotor performance to the non sleep-deprived levels (Lagarde and Batejat, 1995). This result was confirmed by a recent work [START_REF] Wesensten | Effects of modafinil on cognitive performance and alertness during sleep deprivation[END_REF]. Thus, we logically decided in the present study to use the non sleep-deprived animals as controls for post-hoc statistical analysis. The latter result is consistent with the present study, in which the powerful wakening effect of modafinil is clearly evidenced on working memory performance after 10 hrs sleep deprivation. Indeed, after modafinil administration, these animals retrieved their exploratory behavior and a normal working memory performance. Fos is the protein product of the immediate early gene c-fos. In response to a stimulus, one observes first an increase in c-fos mRNA levels, which is soon followed by the synthesis of Fos protein. Fos protein is a transcription factor that, by binding to DNA regulatory regions, can control the expression of many "target" genes. In the CNS, c-fos can be induced by many different stimuli, and it is probably a simpler task to list those stimuli that do not induce c-fos. However, the expression of Fos protein is classically considered as a relevant indicator of neural activity. In sleep-deprived rats, c-fos expression in several brain areas is higher with respect to control animals [START_REF] Cirelli | Sleep deprivation and c-fos expression in the rat brain[END_REF]. Indeed, after a few hours of spontaneous waking or sleep deprivation, c-fos and other IEGs is high in cerebral cortex, hypothalamus, septum, and several thalamic and brainstem nuclei [START_REF] Cirelli | On the functional significance of c-fos induction during the sleep-waking cycle[END_REF]. In addition, [START_REF] Terao | Region-specific changes in immediate early gene expression in response to sleep deprivation and recovery sleep in the mouse brain[END_REF] evidenced a statistically significant increase in c-fos mRNA in mouse cerebral cortex, basal forebrain, thalamus and cerebellum following a 6-hr sleep deprivation. Contrariwise the expression of c-fos in mouse brain is very low after a few hours of sleep/inactivity [START_REF] Basheer | Effects of sleep on wake-induced c-fos expression[END_REF]. Considered together, these results indicate that c-fos expression is associated with waking, but is not proportional to the amount of previous waking [START_REF] Cirelli | On the functional significance of c-fos induction during the sleep-waking cycle[END_REF]. Interestingly, however, it is noteworthy that none of the above studies submitted animals to a behavioural task before Fos analysis.

However, [START_REF] Stone | Depressive behavior in mice due to immune stimulation is accompagned by reduced neural activity in brain regions involved in positively motivated behavior[END_REF] reported that brain areas involved in positively motivated behavior exhibited a decrease in Fos activity, in contrast to brain areas mediating stress effects (e.g.

Paraventricular nucleus

) which exhibited an increase in Fos activity. In the present study, we evidenced significant decreases in c-fos staining after a 10-hr sleep deprivation in all the studied brain areas, as compared to non SD mice. Thus, since our SD model did not induced stress, our results corroborate the general conclusions drawn by [START_REF] Stone | Depressive behavior in mice due to immune stimulation is accompagned by reduced neural activity in brain regions involved in positively motivated behavior[END_REF] concerning the areas which are not involved in stress effects, thus strengthening our hypothesis that stress is not involved in the SD-induced memory impairments. Thus, we can hypothesise that the combination between sleep deprivation on the one hand with the behavioural task (spontaneous alternation) on the other hand might be responsible for these observed decreases in Fos staining. Indeed, this hypothesis is sustained by the work of [START_REF] Célérier | β-CCM enhances retrieval of serial contextual but not of serial spatial memory in mice[END_REF], who already found that the combination of a moderate stress with a cognitive task reduced c-fos expression in the hypothalamus, whereas stress only increased Fos staining in this area, as compared with non-stressed animals. Concerning the effects of modafinil on Fos labelling in brain, modafinil induced marked Fos staining in anterior hypothalamus on cat [START_REF] Lin | Potential brain neuronal targets for amphetamine, methylphenidate, and modafinil-induced wakefulness, evidenced by c-fos immunocytochemistry in the cat[END_REF]1996). Moreover, [START_REF] Engber | Differential patterns of regional c-Fos induction in the rat brain by amphetamine and the novel wakefulness-promoting agent modafinil[END_REF] found by c-Fos immunohistochemistry that the neural targets for modafinil not only include anterior hypothalamus, but also amygdala. These results are consistent with our present work, where Fos staining was previously decreased in anterior hypothalamus and amygdala after sleep deprivation and cognitive task, and then restored by modafinil. However, [START_REF] Scammel | Hypothalamic arousal regions are activated during modafinil-induced wakefulness[END_REF] did not evidence a significant increase in Fos expression in the anterior hypothalamic area after modafinil administration in rat.

Moreover, the debate remains open about a possible relationship between neural activity on the one hand, and memory performance on the other hand. A spatial relationship seems at least to exist, because neural activity variations (evidenced by Fos staining after sleep deprivation followed or not by modafinil administration), partly occurred in the two main brain areas that are involved in memory (i.e. frontal cortex and hippocampus). However, the temporal relationship between memory performance and neural activity changes is not clearly established. In addition, further studies are also needed in order to determine the mechanism of action of modafinil on the recovery of both memory performance and neural activity after sleep deprivation. One could also wonder if a single mechanism of action could explain both the wakening and memory-enhancing effects of modafinil.

A specific contribution of the present study is the evidence, in sleep-deprived mice, of a modafinil-induced enhancement of Fos labelling in amygdala, the main brain area involved in emotions. Indeed, such a finding corroborates our previous study [START_REF] Piérard | Modafinil-induced modulation of working memory and plasma corticosterone in chronically-stressed mice[END_REF] in which we observed an interaction between modafinil, stress and memory. Moreover, as for many memory enhancing drugs, such as beta-CCM [START_REF] Célérier | β-CCM enhances retrieval of serial contextual but not of serial spatial memory in mice[END_REF], an anxiety-like action of modafinil was previously found (unpublished data). In addition, anxiogenic treatments such as cholecystokinin (CCK) fragments increase Fos labelling in several brain regions including amygdala and hypothalamus [START_REF] Netto | Anxiogenic effect of cholecystokinin in the dorsal periaqueductal gray[END_REF]. Thus, the anxiety-like action of modafinil might contribute to its neural activity-recovering effect.

Finally, it will be relevant to extend the present immunohistochemical experiment by the dynamic study, not only of neural activation (using Fos staining), but also as regards neuronal plasticity (by the means of the quantification of MAP2 and synaptophysin protein expression) after sleep deprivation and/or modafinil administration, in the same brain areas as above.

Conclusion:

The current study used a new and original low-stress total sleep deprivation device, that allowed to evidence that a diurnal 10-hr sleep deprivation period induced an impairment of spatial working memory. We observed in the same animals a decrease in Fos protein expression (maybe due to the combination between sleep deprivation and behavioural test), in areas involved in wakesleep cycle regulation (anterior hypothalamus and supraoptic nucleus), but also in memory (frontal cortex and hippocampus) and emotions (amygdala). In a second step, we observed the ability of modafinil (64 but not 32 mg/kg) to restore, after 10 hrs sleep deprivation, both the previously impaired working-memory performance and neural activity.

  3.2.1. On spatial working memory: [Fig. 5] A two-way ANOVA on the overall 4 groups [NSD] (n = 8) ; [SD] (n = 13) ; [SD + M32] (n = 7) and [SD + M64] (n = 8) evidences a significant interaction between groups and interfering trials (trials # 3-6) [F(3,32) = 4.47 ; p < 0.009].

  a 10-hr sleep deprivation period: 3.3.1. On spatial working memory: [Fig. 5] Individual comparisons show that the alternation rate of the [SD + M32] group is not significantly different from [SD] group. On the contrary, the alternation rate of the [SD + M64]

  Figure captions:

Figure 2 :

 2 Figure 2: Behavioural task (spontaneous alternation in a T-maze) used to assess delay-dependent working memory in mice.

Figure 3 :

 3 Figure 3:EEG sample obtained during the occurrence of a short sleep event (14 sec).

Figure 4 :

 4 Figure 4: Mean repartition of sleep events during a diurnal 10-hr sleep deprivation period (n = 7 mice). Left scale: mean number (bars) -Right scale: mean duration (lines) in min.

Figure 5 :

 5 Figure 5: Effects of a 10-hr sleep deprivation period and modafinil on spatial working memory performance. Results are expressed as means ± SEM. NSD: non sleep-deprived ; SD: 10-hr sleep-deprived ; SD + M32: 10-hr sleep-deprived + modafinil 32 mg/kg ; SD + M64: 10-hr sleep-deprived + modafinil 64 mg/kg. ; ***: p < 0.001.

Acknowledgments:

The authors are grateful to Frances Ash (ashberac@free.fr) for language assistance. This research was supported by a grant (Opération n°00 CO 022 -PEA 98 08 12) from Délégation Générale pour l'Armement (DGA/DET/SCET/CEP/SHP, Paris, France) and by the CNRS.


Jorge San Martín

Loredana Smaranda
email: smaranda@dim.uchile.cl

Takéo Takahashi
email: takeo.takahashi@iecn.u-nancy.fr

Convergence of a nite element/ALE method for the Stokes equations in a domain depending on time

We consider the approximation of the unsteady Stokes equations in a time dependent domain when the motion of the domain is given. More precisely, we apply the nite element method to an Arbitrary Lagrangian Eulerian (A.L.E.) formulation of the system. Our main results state the convergence of the solutions of the semi-discretized (with respect to the space variable) and of the fully-discrete problems towards the solutions of the Stokes system.

Introduction

In this work we consider the discretization of a system of partial dierential equations which describes the motion of a viscous incompressible uid in a time dependent domain. More precisely we consider the Stokes system written in a bounded domain Ω t ⊂ R 2 which depends on time t ∈ (0, T ). We want to approximate this system by considering an Arbitrary Lagrangian Eulerian (A.L.E.) formulation for the problem and by using the nite element method.

In many problems and applications one has to work with a uid written in a moving domain. It is generally the case for uid-structure interaction problems like the displacement of shes or of submarines or like the motion of blood in the arteries, etc. Several numerical techniques have been proposed in the literature to overcome the diculty due to the time dependent domain: see, for instance, [START_REF] Masud | A space-time Galerkin/least-squares nite element formulation of the Navier-Stokes equations for moving domain problems[END_REF], [START_REF] Tezduyar | A new strategy for nite element computations involving moving boundaries and interfacesthe deforming-spatial-domain/space-time procedure. I. The concept and the preliminary numerical tests[END_REF], [START_REF] Tezduyar | A new strategy for nite element computations involving moving boundaries and interfacesthe deforming-spatial-domain/space-time procedure. II. Computation of free-surface ows, two-liquid ows, and ows with drifting cylinders[END_REF], [START_REF] Chang | A level set formulation of Eulerian interface capturing methods for incompressible uid ows[END_REF], [START_REF] Glowinski | A ctitious domain method for external incompressible viscous ow modeled by Navier-Stokes equations[END_REF], [START_REF]A ctitious domain method for unsteady incompressible viscous ow modelled by Navier-Stokes equations[END_REF], [START_REF] Peskin | Numerical analysis of blood ow in the heart[END_REF], [START_REF] San Martín | Convergence of the Lagrange-Galerkin method for the equations modelling the motion of a uid-rigid system[END_REF], [START_REF] Bertoluzza | The fat boundary method: semi-discrete scheme and some numerical experiments[END_REF], [START_REF] Maury | A fat boundary method for the Poisson problem in a domain with holes[END_REF]. Here we consider the Arbitrary Lagrangian Eulerian (A.L.E.) method which main idea consists in moving in a convenient way the mesh in order to follow the motion of the domain, instead of re-meshing at each step time (which leads to a too expensive computation). If the deformation of the domain is not too important, it is possible to keep the regularity properties of the initial grid. This method have been proposed and studied by many authors: [START_REF] Donea | Arbitrary Lagrangian Eulerian Methods[END_REF], [START_REF] Farhat | Mixed explicit/implicit time integration of coupled aeroelastic problems: three-eld formulation, geometric conservation and distributed solution[END_REF], [START_REF] Guillard | On the signicance of the geometric conservation law for ow computations on moving meshes[END_REF], [START_REF] Formaggia | A stability analysis for the arbitrary Lagrangian Eulerian formulation with nite elements[END_REF], [START_REF] Nobile | Numerical approximation of uid-structure interaction problems with application to haemodynamics[END_REF], [START_REF] Gastaldi | A priori error estimates for the arbitrary Lagrangian Eulerian formulation with nite elements[END_REF], [START_REF] Maury | Characteristics ALE method for the unsteady 3D Navier-Stokes equations with a free surface[END_REF], [START_REF] Duarte | Arbitrary Lagrangian-Eulerian method for Navier-Stokes equations with moving boundaries[END_REF], [START_REF] Girault | One time-step nite element discretization of the equation of motion of two-uid ows[END_REF].

For many uid-structure interaction problems, the motion of the domain is time dependent but is also an unknown of the problem and the equations for the uid have to be coupled with some equations for the structure. For instance, if we deal with the motion of rigid bodies into a viscous incompressible uid, the problem can be modeled by the coupling between the Navier-Stokes equations (corresponding to the uid part) and ordinary dierential equations (corresponding to the rigid bodies). The problem could even be 1 more complicated if the structure is deformable and although many authors (see, for instance, [START_REF] Beirão | On the existence of strong solutions to a coupled uid-structure evolution problem[END_REF], [START_REF] Boulakia | Existence of weak solutions for an interaction problem between an elastic structure and a compressible viscous uid[END_REF], [START_REF] Desjardins | Weak solutions for a uid-elastic structure interaction model[END_REF], [START_REF] Chambolle | Existence of weak solutions for the unsteady interaction of a viscous uid with an elastic plate[END_REF]) have tackled the well-posedness of such systems, there are still many open questions (even for deriving a model with good properties). In the present paper, we tackle the problem when the motion of the domain is given. Moreover, to split the diculties, we focus on the Stokes equations, neglecting the non linear term of the Navier-Stokes system. Despite these hypotheses, the problem remains complicated since we have to consider a mixed formulation in a time dependent domain, which is completely new with respect to the recent literature.

Let us briey recall some reference about the numerical convergence for the Stokes/Navier-Stokes equations and the uid-structure interaction problems. In the case of a xed domain, and for the Navier-Stokes equations, the Lagrange-Galerkin method has been proposed and analyzed in [START_REF] Pironneau | On the transport-diusion algorithm and its applications to the Navier-Stokes equations[END_REF]. In [START_REF] Süli | Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations[END_REF], the author has proved optimal error estimates for the Lagrange-Galerkin mixed nite element approximation of Navier-Stokes equations in a velocity/pressure formulation. We also mention the work of Achdou and Guermond [START_REF] Achdou | Convergence analysis of a nite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations[END_REF], where convergence analysis of a nite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations is done. In the case where the domain is time dependent but given, the convergence analysis for the ALE method has been considered by [START_REF] Formaggia | A stability analysis for the arbitrary Lagrangian Eulerian formulation with nite elements[END_REF], [START_REF] Nobile | Numerical approximation of uid-structure interaction problems with application to haemodynamics[END_REF], [START_REF] Gastaldi | A priori error estimates for the arbitrary Lagrangian Eulerian formulation with nite elements[END_REF], in the case of the advection-diusion equation instead of the Stokes or the Navier-Stokes equations. Finally, when the domain is time dependent but unknown, few results exist in the literature: Grandmont, Guimet and Maday (in [START_REF] Grandmont | Numerical analysis of some decoupling techniques for the approximation of the unsteady uid structure interaction[END_REF]) deal with the case of one dimensional problem discretized by using the ALE formulation. In [START_REF] San Martín | Convergence of the Lagrange-Galerkin method for the equations modelling the motion of a uid-rigid system[END_REF] the authors have proved the convergence of a numerical method based on the use of characteristics and on nite elements with a xed mesh for a two dimensional uid-rigid body problem.

Let us describe more precisely our problem. For a given T > 0, and for each t ∈ [0, T ], we consider a bounded polyhedral convex domain Ω t in R 2 . We set

Q T = (x, t) ∈ R 3 | x ∈ Ω t , t ∈ (0, T ) .
The Stokes system in the domain Ω t , t ∈ (0, T ) can be written as follows:

         ∂u ∂t -ν∆u + ∇p = f in Q T , div u = 0 in Q T , u = 0
on ∂Ω t , t ∈ (0, T ), u(0) = u 0 in Ω 0 .

(1. [START_REF] Achdou | Convergence analysis of a nite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations[END_REF] In these equations, u = (u 1 , u 2 ) is the velocity of the uid, its density is assumed to be equal to 1, ν > 0 is its constant kinematic viscosity and p is its pressure; f = (f 1 , f 2 ) represents a density of body forces per unit mass (for instance, gravity).

It can be proved that the system (1.1) is well-posed provided that Q T and the data (f and u 0 ) are smooth enough. The diculty in this proof, which comes from the fact that the domain is moving on time, has been overcome by several works. We mention, among others, the paper of Ôtani and Yamada [START_REF] Ôtani | On the Navier-Stokes equations in noncylindrical domains: an approach by the subdierential operator theory[END_REF] and the work of Inoue and Wakimoto [START_REF] Inoue | On existence of solutions of the Navier-Stokes equation in a time dependent domain[END_REF]. In the last one, the equations (1.1) are recast on a cylindrical space time domain introducing a suitable dieomorphism. A result of existence of a weak solution is obtained also in Salvi [START_REF]The exterior nonstationary problem for the Navier-Stokes equations ireference [13]n regions with moving boundaries[END_REF], [START_REF] Salvi | On the existence of weak solutions of a nonlinear mixed problem for the Navier-Stokes equations in a time dependent domain[END_REF] through an elliptic regularization, under weaker hypotheses on the regularity of the domain boundary than in the previously cited paper.

The paper is organized as follows. In the next section we deal with the ALE formulation of the Stokes system and we state our main results. The rst result given in Theorem 2.1 consists in the convergence of a semi-discretization scheme with respect to the space variable and the second one (Theorem 2.2) states an error estimate for a fully-discrete formulation. Section 3 is devoted to some preliminary results useful to prove our main theorems. In Section 4 we introduce the projections on the nite element spaces and we prove some estimates for their time derivative on the ALE frame. The fth section is devoted to the proof of the rst main result and nally, in Section 6 we prove the second main result.

2 Statement of the main results

2.1

The ALE formulation of the Stokes equations Let rst give some assumptions on the non cylindrical domain Q T . We assume that there exists a mapping X ∈ H 1 (0, T ; W 2,∞ (Ω 0 ) 2 ) such that for each t ∈ (0, T ), the mapping

X t : Ω 0 -→ Ω t , y -→ X(y, t), (2.1)
is invertible and X -1

t ∈ W 1,∞ (Ω t ) 2 .
In the literature, y ∈ Ω 0 is called the ALE coordinate, and x ∈ Ω t the spatial (or Eulerian) coordinate.

Using the transformation X, we can write the ALE formulation of (1.1). To achieve this, we introduce the following notation: rst, we denote by w the domain velocity, which is dened by

w : Q T -→ R 2 , w(x, t) = ∂X ∂t (X -1 t (x), t).
dv dt Y : Q T -→ R, (x, t) -→ dv dt Y (x, t) = ∂v ∂t (x, t) + w(x, t) • ∇v(x, t). (2.3)
Using this denition, we obtain that the Stokes system (1.1) is equivalent to the following system (ALE formulation of (1.1)):

           du dt Y -ν∆u + ∇p -(w • ∇) u = f in Q T , div u = 0 in Q T , u = 0 on ∂Ω t , t ∈ (0, T ), u(0) = u 0 in Ω 0 . (2.4)
It may be noticed that the main dierence with the original formulation (1.1) is the appearance of the convective-type term due to the domain movement.

In order to write the ALE weak formulation of problem (2.4) we need some results on the time derivatives of integrals on moving domains. This kind of results will be developed in details in Section 3. Using these results, we get the following mixed weak formulation:

Find u : Q T → R 2 and p : Q T → R such that for each t ∈ (0, T ), u(•, t) ∈ H 1 0 (Ω t ) 2 , p(•, t) ∈ L 2 0 (Ω t
) and the following system holds:

                           d dt Ωt u • (v • X -1 t ) dx + Ωt ∇u : ∇(v • X -1 t ) dx - Ω t div (w ⊗ u) • (v • X -1 t ) dx - Ωt p div (v • X -1 t ) dx = Ωt f • (v • X -1 t ) dx ∀v ∈ H 1 0 (Ω 0 ) 2 , Ω t (q • X -1 t )div u dx = 0 ∀q ∈ L 2 (Ω 0 ), u(•, 0) = u 0 (•) in Ω 0 , (2.5)
where for any open set Ω ⊂ R 2 , we have denoted by L 2 0 (Ω) the classical pressure space, that is:

L 2 0 (Ω) = f ∈ L 2 (Ω) Ω f (x) dx = 0 .
Let us also introduce the classical space of free divergence elds associated to Stokes problem, dened by

H 1 0,σ (Ω) = u ∈ H 1 0 (Ω) 2 | div u = 0 .
Since we deal with the mixed formulation (2.5), it is natural to assume the following uniform inf-sup condition:

inf

p∈L 2 0 (Ωt) sup v∈H 1 0 (Ωt) 2 Ωt p div v dx v H 1 (Ωt) 2 p L 2 (Ωt) β, (2.6)
where β is a positive constant which does not depend on time. The inf-sup condition was introduced independently by Babu²ka [START_REF] Babu²ka | The nite element method with Lagrangian multipliers[END_REF] and Bezzi [START_REF] Brezzi | On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers[END_REF]. Notice that a sucient condition to guarantee (2.6) is that the deformation of Ω t is small. More precisely, there exists a constant α > 0 depending only on Ω 0 such that if

X -Id L ∞ (Ω 0 ×(0,T )) 2 + ∇X -Id L ∞ (Ω 0 ×(0,T )) 4 < α, (2.7)
then (2.6) holds true. It is important to remark that the assumption (2.7) is quite natural: indeed, in practice, the ALE formulation can not be used to discretize a problem when the deformation are too big and it is usually necessary to re-mesh the domain to preserve the regularity of the mesh (see, [START_REF] Maury | Direct simulations of 2D uid-particle ows in biperiodic domains[END_REF] for instance)

2.2 Semi-discretization scheme and statement of the rst main result

In order to discretize our problem with respect to the space variable, we introduce two nite element spaces of Hood-Taylor type; these spaces depend on time since our problem is written on the domain Ω t .

Let h denote a discretization parameter, with 0 < h < 1. At initial time t = 0, we consider a quasiuniform triangulation T h,0 of Ω 0 , as dened, for instance, in Brenner and Scott [6, p.106]. We also assume that there is no triangle of T h,0 with two edges on ∂Ω 0 . These assumptions on T h,0 will be assumed throughout this paper.

For any t ∈ [0, T ], we consider a discretization of the mapping X t by means of piecewise linear Lagrangian nite elements, denoted by X h,t :

X h,t : Ω 0 -→ Ω t , y -→ X h,t (y).
We assume that X h,t is smooth and invertible. Let T h,t be the image of T h,0 under the discrete ALE mapping X h,t .

We associate to this triangulation two classical approximation spaces used in the mixed nite element methods for the Stokes system. The rst space, classically used for the approximation of the velocity eld in the mixed statement of the Stokes system, is denoted by W h,t and is composed with the P 2 -nite elements associated to T h,t . More precisely:

W h,t = v h ∈ H 1 0 (Ω t ) | v h| K ∈ P 2 (K) ∀K ∈ T h,t ,
where P n (K) is the set of polynomials on K of degree less than or equal to n.

The second space, classically used for the approximation of the pressure in mixed formulations of the Stokes system, is denoted by M h,t and is composed with the P 1 -nite elements associated to T h,t , that is,

M h,t = q h ∈ H 1 (Ω t ) | q h| K ∈ P 1 (K) ∀K ∈ T h,t .
We also consider the space

M 0 h,t = M h,t ∩ L 2 0 (Ω t ).
Since Ω 0 is a polyhedral convex domain and X h,t is piecewise linear and smooth, we can characterize the spaces W h,t and M h,t as follows:

W h,t = v h • X -1 h,t | v h ∈ W h,0 , (2.8)
M h,t = q h • X -1 h,t | q h ∈ M h,0 .
(2.9)

As in the previous subsection, we consider w h the velocity eld associated to the discrete ALE mapping:

w h (x, t) = ∂X h,t ∂t X -1 h,t (x) .
Using this discrete velocity eld, we can introduce the time derivative on the discrete ALE frame as follows: for any v :

Q T -→ R smooth enough, we dene dv dt h Y (x, t) = ∂v ∂t (x, t) + w h (x, t) • ∇v(x, t).
(2.10)

Now, using the weak ALE formulation (2.5) and the denitions above, we can derive a semi-discrete version of our problem. For any h ∈ (0, 1) we denote by u h and p h the solution of the following problem: Find u h and p h such that u h (•, 0) = u h,0 and for any t ∈ (0, T ), u h (•, t) ∈ (W h,t ) 2 , p h (•, t) ∈ M 0 h,t and the following system holds

                     d dt Ω t u h • v h • X -1 h,t dx + ν Ωt ∇u h : ∇ v h • X -1 h,t dx - Ω t div (w h ⊗ u h ) • v h • X -1 h,t dx - Ω t p h div v h • X -1 h,t dx = I h,t f (t) • v h • X -1 h,t ∀v h ∈ (W h,0 ) 2 , Ωt q h • X -1 h,t div u h dx = 0 ∀q h ∈ M h,0 , (2.11)
where u h,0 is a nite element approximation of the initial data u 0 . In the third line we have used the notation I h,t (F ) to denote a numerical quadrature formula for the integral

Ω t F (x) dx.
In the rest of paper, we assume that the quadrature formula is exact for the continuous functions in Ω t , whose restriction of each triangle is polynomial of degree less than or equal to 4. Using this fact, each integral of the above numerical scheme can be replaced by the numerical integration formula.

To get the convergence of the numerical scheme, it essential to assume that the discrete ALE mapping X h approximates X in some sense. More precisely, we assume that the following error estimate holds true.

X t -X h,t L ∞ (Ω 0 ) 2 + h ∇(X t -X h,t ) L ∞ (Ω 0 ) 4 ≤ C h 2 | ln h| X t W 2,∞ (Ω 0 ) 2 .
(2.12)

For more details about the construction of a mapping X h satisfying such an estimate, we refer the reader to Gastaldi [START_REF] Gastaldi | A priori error estimates for the arbitrary Lagrangian Eulerian formulation with nite elements[END_REF]. We can notice that if we assume w(t) ∈ W 2,∞ (Ω t ) 2 , then the following error estimate on the domain velocity holds true (for more details, see Gastaldi [START_REF] Gastaldi | A priori error estimates for the arbitrary Lagrangian Eulerian formulation with nite elements[END_REF]): for all t ∈ (0, T ),

w(t) -w h (t) L ∞ (Ωt) 2 + h ∇(w(t) -w h (t)) L ∞ (Ωt) 4 ≤ Ch 2 |lnh| w(t) W 2,∞ (Ωt) 2 .
(2.13)

The other important hypothesis to obtain the convergence of our scheme is that the triangulation T h,t remains non-degenerate with the time (see [6, pp.106-107]): we assume that there exists ρ > 0 such that

diam B K ≥ ρh diam K ∀K ∈ T h,t (2.14)
for all t ∈ [0, T ] and for all h ∈ (0, 1], where B K is the largest disk contained in K. In practice, this hypothesis holds only for a small time interval, especially when one deals with great deformations. If we assume that T h,0 is non-degenerate, that the deformation is small enough (see (2.7)) and that the approximation X h is close to X (see (2.12)), then for h small enough, we can prove that (2.14) holds true.

We are now in position to state the rst main result of the paper:

Theorem 2.1. Suppose that the above assumptions on T h,t and on X h hold true and that (2.6) is satised.

Let also assume that the solution (u, p) of the problem (2.4) and the data w, f satisfy the following properties:

u ∈ L ∞ (0, T ; H 3 (Ω t ) 2 ∩ H 1 0,σ (Ω t )), du dt Y ∈ L 2 (0, T ; H 2 (Ω t ) 2 ), u(0) ∈ H 3 (Ω t ) 2 , p ∈ L ∞ (0, T ; H 2 (Ω t ) ∩ L 2 0 (Ω t )), dp dt Y ∈ L 2 (0, T ; H 1 (Ω t )), p(0) ∈ H 2 (Ω t ), w ∈ L ∞ (0, T ; W 2,∞ (Ω t ) 2 ), f ∈ L 2 (0, T ; W 2,q (Ω t ) 2 ), for some q > 2.            (2.15)
Then there exists a constant C > 0, independent of h, such that the solution (u h , p h ) of the semidiscretization problem (2.11) satises

u -u h 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + ν ∇(u -u h ) 2 L 2 (0,T ;L 2 (Ωt) 4 ) ≤ u 0 -u h,0 2 L 2 (Ω 0 ) 2 + Ch 2 | ln h| 2 f 2 L 2 (0,T ;W 2,q (Ωt) 2 ) + u 2 L ∞ (0,T ;H 3 (Ωt) 2 ) + du dt Y 2 L 2 (0,T ;H 2 (Ωt) 2 ) + p 2 L ∞ (0,T ;H 2 (Ωt)) + dp dt Y 2 L 2 (0,T ;H 1 (Ωt))
.

(2.16)

The fully-discrete formulation and statement of the second main result

In order to discretize our problem with respect to the time variable, let us denote by ∆t > 0 the time step and t n = n∆t, for n = 0, . . . , N, where N is such that t N ≤ T and t N +1 > T.

In the fully-discrete problem, we will consider a piecewise linear interpolation in time of the domain deformation. Thus, the domain velocity is constant on each interval (t n , t n+1 ) and at time t = t n+1 is given by:

w * h,n,n+1 (x) = 1 ∆t x -X h,tn X -1 h,t n+1 (x) ∀x ∈ Ω t n+1 ,
for all n ∈ {0, . . . , N -1}.

With the above denitions, we can introduce the fully-discrete problem, using an implicit Euler scheme, as follows:

Find {u n h } and {p n h } such that u 0 h = u h,0 and for any n = 0, . . . , N -1, one has that

u n+1 h ∈ (W h,t n+1 ) 2 , p n+1 h ∈ M 0
h,t n+1 and the following system holds:

                                       Ω t n+1 u n+1 h • (v h • X -1 h,t n+1 ) dx - Ω tn u n h • (v h • X -1 h,t n ) dx +ν∆t Ωt n+1 ∇u n+1 h : ∇(v h • X -1 h,t n+1 ) dx -∆t Ωt n+1 div w * h,n,n+1 ⊗ u n+1 h • (v h • X -1 h,t n+1 ) dx -∆t Ωt n+1 p n+1 h div (v h • X -1 h,t n+1 ) dx = ∆t I h,t n+1 f (t n+1 ) • (v h • X -1 h,t n+1 ) ∀v h ∈ (W h,0 ) 2 , Ω t n+1 (q h • X -1 h,t n+1 ) div u n+1 h dx = 0 ∀q h ∈ M h,0 .
(2.17)

In the sequel, we state the second main result of this paper, which gives the error estimate in the approach given by the ALE method for the Stokes problem in time depending domain. More precisely, we have the following theorem:

Theorem 2.2. Suppose that the assumptions of Theorem 2.1 hold true. Let also assume that

∂ 2 X h ∂t 2 ∈ L ∞ (0, T ; L ∞ (Ω 0 ) 2 ) and df dt Y ∈ L 2 (0, T ; L 2 (Ω t ) 2 ).
(

Then, there exists a positive constant C, independent of h and ∆t, such that for all suciently small ∆t and h, we have the following error estimate:

u(t n+1 ) -u n+1 h 2 L 2 (Ωt n+1 ) 2 + ν ∆t n+1 i=1 ∇ u(t i ) -u i h 2 L 2 (Ωt i ) 4 ≤ u 0 -u h,0 2 L 2 (Ω 0 ) 2 + C h 4 ∆t 2 + h 4 u 2 L ∞ (0,T ;H 3 (Ωt) 2 ) + p 2 L ∞ (0,T ;H 2 (Ωt)) + C∆t 2 sup s∈(0,T ) ∂ 2 X h ∂s 2 (s) 2 L ∞ (Ω 0 ) 2 u 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + C∆t h 4 n+1 i=1 f (t i ) 2 W 2,q (Ωt i ) 2 + C∆t 2 t n+1 0 du dt Y (t) 2 H 1 (Ωt) 2 + u(t) 2 H 2 (Ωt) 2 + dp dt Y (t) 2 L 2 (Ωt) + p(t) 2 H 1 (Ωt) + df dt Y (t) 2 L 2 (Ωt) 2 + f (t) 2 H 1 (Ωt) 2 dt. (2.19)
Remark 2.3. In particular, if there exists a xed constant C 0 > 0 such that h ≤ C 0 ∆t and u(0) -

u h,0 L 2 (Ω 0 ) 2 ≤ C 0 h, we have that u(t n+1 ) -u n+1 h 2 L 2 (Ωt n+1 ) 2 + ν ∆t n+1 i=1 ∇ u(t i ) -u i h 2 L 2 (Ωt i ) 4 ≤ C∆t 2 .
Remark 2.4. Let us observe that the condition h ≤ C 0 ∆t is quite natural for the convergence of mixed schemes. For instance, in [START_REF] Pironneau | On the transport-diusion algorithm and its applications to the Navier-Stokes equations[END_REF] the convergence is obtained for h ≤ C 0 ∆t and in [START_REF] Süli | Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations[END_REF] for h 2 ≤ C 0 ∆t ≤ C 1 h σ and σ > 1/2 (with h and ∆t small enough).

Remark 2.5. The regularity assumption (2.18) on X h is quite natural in the case of a time depending operator, in order to obtain the fully error estimate (2.19) given above in the Theorem 2.2. If we use the construction of X h and its continuous counterpart X, given in Gastaldi [START_REF] Gastaldi | A priori error estimates for the arbitrary Lagrangian Eulerian formulation with nite elements[END_REF], it is clear that this regularity with respect to t is strictly related with the displacement of the boundary.

Preliminary results

This section is devoted to some preliminary results which will be useful to prove Theorem 2.1 and Theorem 2.2. These results are either easy to prove or classical and, for this reason, we shall omit all the proofs in what follows.

Let us rst recall the following classical result (see, for instance [22, pp.19-20]). In the context of ALE formulations, this result has been also presented in [START_REF] Formaggia | A stability analysis for the arbitrary Lagrangian Eulerian formulation with nite elements[END_REF].

Proposition 3.1. Consider Ω 1 and Ω 2 two bounded open subsets of R 2 and assume that X ∈ W 1,∞ (Ω 1 ). Suppose also that X : Ω 1 → Ω 2 is invertible and such that X -1 ∈ W 1,∞ (Ω 2 ). Then for any u ∈ H 1 (Ω 2 ) we have that u • X ∈ H 1 (Ω 1 ).
This proposition justify the mixed formulation (2.5) and will be used throughout the paper.

Since we have to deal with integrals on moving domain in this problem, we give also some useful formulas for the time derivative of integrals on moving domains. First of all, we recall the Reynolds transport formula, that is, let ψ(x, t) be a smooth function dened on Q T . Then for any open subdomain

V t ⊆ Ω t such that V t = X t (V 0 ) with V 0 ⊆ Ω 0 , we have that d dt Vt ψ(x, t) dx = Vt ∂ψ ∂t + ∇ψ • w + ψ div w dx = Vt dψ dt Y + ψ div w dx
(see, for instance, Gurtin [START_REF] Gurtin | An introduction to continuum mechanics[END_REF]).

Furthermore, since for any χ : Ω 0 → R 2 we have that

d dt χ • X -1 t Y
= 0, it is not dicult to prove the following lemma, which is a consequence of the above formula.

Lemma 3.2. Let assume that ϕ :

Q T → R 2 , ψ : Q T → R and χ : Ω 0 → R 2 are smooth functions. Then
we have the following relations:

d dt Ωt χ • X -1 t • ϕdx = Ωt χ • X -1 t • dϕ dt Y + ϕdiv w dx, (3.1)
d dt Ωt ∇ϕ : ∇ χ • X -1 t dx = Ωt ∇ dϕ dt Y : ∇ χ • X -1 t + ∇ϕ : ∇ χ • X -1 t div w -∇w + ∇w T ∇ϕ : ∇ χ • X -1 t dx, (3.2) d dt Ω t χ • X -1 t div ϕ dx = Ω t χ • X -1 t div dϕ dt Y + χ • X -1 t div ϕdiv w -χ • X -1 t ∇w : ∇ϕ T dx, (3.3) d dt Ωt ψdiv χ • X -1 t dx = Ωt dψ dt Y div χ • X -1 t + ψdiv χ • X -1 t div w -ψ ∇w : ∇ χ • X -1 t T dx. (3.4)
It is well-known (see, for instance, [START_REF] Girault | Finite element methods for Navier-Stokes equations[END_REF]) that the mixed formulation (2.11) is a well-posed problem, provided that the spaces W h,t , M 0 h,t and the bilinear form

b(p h , v h ) := Ωt p h div v h dx
satisfy the Brezzi-Babu²ka (inf-sup) condition. The fact that this inf-sup condition is satised in our case, at each time t ∈ (0, T ), follows from the choice of the nite element used. That is, at each time t ∈ (0, T ), there exists a positive constant β t such that

inf p h ∈M 0 h,t sup v h ∈(W h,t ) 2 Ωt p h div v h dx v h H 1 (Ωt) 2 p h L 2 (Ωt) β t .
In fact, if h is small enough, we can choose a constant β * independent of t instead of β t in the above inequality. More precisely, we have the following result. Theorem 3.3. Assume that (2.6) and (2.14) hold true. Then there exist two positive constant h * and β * such that for all t ∈ (0, T ) and for all h ∈ (0, h * ), inf

p h ∈M 0 h,t sup v h ∈(W h,t ) 2 Ωt p h div v h dx v h H 1 (Ω t ) 2 p h L 2 (Ω t ) β * . (3.5)
This theorem can be easily proved by using (2.14) and (2.6) and by following the proof of Theorem 10.6.6 in [START_REF] Brenner | The mathematical theory of nite element methods[END_REF]. Therefore, we omit the proof of the preceding theorem. [START_REF] Bertoluzza | The fat boundary method: semi-discrete scheme and some numerical experiments[END_REF] Estimates of the projection on the nite element spaces One of the key ingredient in the proof of our convergence results is the introduction of a projection on the nite element space (W h,t ) 2 × M 0 h,t of the exact problem solution

(u, p) ∈ H s+1 (Ω t ) 2 ∩ H 1 0 (Ω t ) 2 × H s (Ω t ) ∩ L 2 0 (Ω t )
(with s a real number s ≥ 1).

Proposition 4.1. Suppose that s ≥ 1 is a real number. If u(t) ∈ H s+1 (Ω t ) 2 ∩ H 1 0 (Ω t ) 2 and p(t) ∈ H s (Ω t ) ∩ L 2 0 (Ω t )
, then there exists an unique couple (U(t),

P (t)) in (W h,t ) 2 × M 0 h,t such that        ν Ωt ∇ (U(t) -u(t)) : ∇v h dx - Ωt (P (t) -p(t)) div v h dx = 0 ∀v h ∈ (W h,t ) 2 , Ω t q h div (U(t) -u(t)) dx = 0 ∀q h ∈ M h,t . (4.1)
Moreover, there exists a positive constant C > 0, independent of h and t, such that

u(t) -U(t) H 1 (Ω t ) 2 + p(t) -P (t) L 2 (Ωt) ≤ Ch r u(t) H r+1 (Ωt) 2 + p(t) H r (Ω t ) , (4.2)
for all r such that 1 ≤ r ≤ min(2, s).

The proof of this Proposition is a direct consequence of Theorem 1.1 from Girault and Raviart (see [18, p.114]) and of Theorem 3.3. Remark 4.2. Due to Proposition 3.1, the problem (4.1) is equivalent to the following one:

               ν Ωt ∇ (U(t) -u(t)) : ∇ v h • X -1 h,t dx - Ωt (P (t) -p(t)) div v h • X -1 h,t dx = 0 ∀v h ∈ (W h,0 ) 2 , Ωt q h • X -1 h,t div (U(t) -u(t)) dx = 0 ∀q h ∈ M h,0 . (4.3)
In order to prove our main results, we need some estimates of the time derivatives on the ALE frame for the projections introduced above. More precisely, we get the following theorem:

Theorem 4.3. Assume that u : Q T -→ R 2 , p : Q T -→ R satisfy u(t) ∈ H 3 (Ω t ) 2 ∩ H 1 0,σ (Ω t ), p(t) ∈ H 2 (Ω t ) ∩ L 2 0 (Ω t ),
for all t ∈ (0, T ).

Let consider the projection (U(t), P (t)) onto (W h,t ) 2 × M 0 h,t of (u(t), p(t)), dened in Proposition 4.1. We assume that

w(t) ∈ W 2,∞ (Ω t ) 2 , du dt Y (t) ∈ H 2 (Ω t ) 2 , dp dt Y (t) ∈ H 1 (Ω t ). (4.4)
Then there exists a positive constant C, independent of h, such that

du dt h Y (t) - dU dt h Y (t) H 1 (Ωt) 2 + dp dt h Y (t) - dP dt h Y (t) L 2 (Ωt) ≤ Ch| ln h| u(t) H 3 (Ωt) 2 + du dt Y (t) H 2 (Ωt) 2 + p(t) H 2 (Ωt) + dp dt Y (t) H 1 (Ωt)
. (4.5)

Proof. Using (3.2)(3.4) we dierentiate with respect to t the both equations of (4.3), then we obtain: for

all v h ∈ (W h,0 ) 2 and q h ∈ M h,0 , ν Ωt ∇ dU dt h Y (t) - du dt h Y (t) : ∇ v h • X -1 h,t dx - Ω t dP dt h Y (t) - dp dt h Y (t) div v h • X -1 h,t dx = -ν Ωt ∇ (U(t) -u(t)) : ∇ v h • X -1 h,t div w h (t) dx + ν Ωt ∇w h (t) + ∇w h (t) T ∇ (U(t) -u(t)) : ∇ v h • X -1 h,t dx + Ωt (P (t) -p(t)) div v h • X -1 h,t div w h (t) dx - Ω t (P (t) -p(t)) ∇w h (t) : ∇ v h • X -1 h,t T dx, (4.6a) and Ωt q h • X -1 h,t div dU dt h Y (t) - du dt h Y (t) dx = - Ωt q h • X -1 h,t div (U(t) -u(t)) div w h (t) dx + Ω t q h • X -1 h,t ∇w h (t) : ∇ (U(t) -u(t)) T dx. (4.6b)
Now, we recall that

du dt h Y (t) = du dt Y (t) + ((w h (t) -w(t)) • ∇) u(t), (4.7
)

dp dt h Y (t) = dp dt Y (t) + (w h (t) -w(t)) • ∇p(t), (4.8)
therefore, the system (4.6a)(4.6b) can be written as follows: for all v h ∈ (W h,0 ) 2 and q h ∈ M h,0 , ν

Ω t ∇ dU dt h Y (t) - du dt Y (t) : ∇ v h • X -1 h,t dx - Ωt dP dt h Y (t) - dp dt Y (t) div v h • X -1 h,t dx = -ν Ω t ∇ (U(t) -u(t)) : ∇ v h • X -1 h,t div w h (t) dx + ν Ωt ∇w h (t) + ∇w h (t) T ∇ (U(t) -u(t)) : ∇ v h • X -1 h,t dx + Ω t (P (t) -p(t)) div v h • X -1 h,t div w h (t) dx - Ωt (P (t) -p(t)) ∇w h (t) : ∇ v h • X -1 h,t T dx + ν Ωt ∇ ((w h (t) -w(t)) • ∇) u(t) : ∇ v h • X -1 h,t dx - Ωt (w h (t) -w(t)) • ∇p(t) div v h • X -1 h,t dx (4.9a) and Ωt q h • X -1 h,t div dU dt h Y (t) - du dt Y (t) dx = - Ω t q h • X -1 h,t div (U(t) -u(t)) div w h (t) dx + Ω t q h • X -1 h,t ∇w h (t) : ∇ (U(t) -u(t)) T dx + Ωt q h • X -1 h,t div ((w h (t) -w(t)) • ∇) u(t) dx. (4.9b)
On the other hand, we have that (t), p1 (t) , which are solutions of the following well-dened problem:

du dt Y (t) ∈ H 2 (Ω t ) 2 ∩ H 1 0 (Ω t )
               ν Ωt ∇ U 1 (t) - du dt Y (t) : ∇ v h • X -1 h,t dx - Ωt P 1 (t) - dp dt Y (t) + λ div v h • X -1 h,t dx = 0 ∀v h ∈ (W h,0 ) 2 , Ωt q h • X -1 h,t div U 1 (t) - du dt Y (t) dx = 0 ∀q h ∈ M h,0 . (4.10)
From Proposition 4.1 and Proposition 3.1, we have that

U 1 (t) - du dt Y (t) H 1 (Ωt) 2 + P 1 (t) - dp dt Y (t) + λ L 2 (Ωt) ≤ Ch du dt Y (t) H 2 (Ω t ) 2 + dp dt Y (t) H 1 (Ω t )
. (4.11)

Subtracting (4.10) from the system obtained by (4.9a) and (4.9b), we get the following problem: for all v h ∈ (W h,0 ) 2 and q h ∈ M h,0 , ν

Ω t ∇ dU dt h Y (t) -U 1 (t) : ∇ v h • X -1 h,t dx - Ωt dP dt h Y (t) -P 1 (t) div v h • X -1 h,t dx = -ν Ωt ∇ (U(t) -u(t)) : ∇ v h • X -1 h,t div w h (t) dx + ν Ωt ∇w h (t) + ∇w h (t) T ∇ (U(t) -u(t)) : ∇ v h • X -1 h,t dx + Ωt (P (t) -p(t)) div v h • X -1 h,t div w h (t) dx - Ωt (P (t) -p(t)) ∇w h (t) : ∇ v h • X -1 h,t T dx + ν Ω t ∇ ((w h (t) -w(t)) • ∇) u(t) : ∇ v h • X -1 h,t dx - Ωt (w h (t) -w(t)) • ∇p(t)div v h • X -1 h,t dx (4.12a) and Ωt q h • X -1 h,t div dU dt h Y (t) -U 1 (t) dx = - Ωt q h • X -1 h,t div (U(t) -u(t)) div w h (t) dx + Ωt q h • X -1 h,t ∇w h (t) : ∇ (U(t) -u(t)) T dx + Ω t q h • X -1 h,t div ((w h (t) -w(t)) • ∇) u(t)
dU dt h Y (t) -U 1 (t) H 1 (Ωt) 2 + dP dt h Y (t) -λ h -P 1 (t) L 2 (Ωt) ≤ C ∇w h (t) L ∞ (Ω t ) 4 ∇ (U(t) -u(t)) L 2 (Ω t ) 4 + P (t) -p(t) L 2 (Ω t ) + C w h (t) -w(t) W 1,∞ (Ωt) 2 u(t) H 2 (Ωt) 2 + p(t) H 1 (Ωt) , (4.13)
where the constant C > 0 is independent of h and t. Therefore, using (2.13) and (4.2) (with s = r = 2), the estimate (4.13) becomes then by dierentiating with respect to t we get

dU dt h Y (t) -U 1 (t) H 1 (Ωt) 2 + dP dt h Y (t) -λ h -P 1 (t) L 2 (Ωt) ≤ Ch| ln h| w(t) W 2,∞ (Ω t ) 2 u(t) H 3 (Ω t ) 2 + p(t) H 2 (Ω t ) . ( 4
dU dt h Y (t) - du dt Y (t) H 1 (Ω t ) 2 + dP dt h Y (t) - dp dt Y (t) L 2 (Ω t ) ≤ Ch| ln h| u(t) H 3 (Ωt) 2 + du dt Y (t) H 2 (Ωt) 2 + p(t) H 2 (Ω t ) + dp dt Y (t) H 1 (Ωt) + C|λ -λ h |.
λ = - 1 |Ω t | Ωt p(t)div w(t) dx and λ h = - 1 |Ω t | Ωt P (t)div w h (t) dx.
Hence,

|λ -λ h | ≤ 1 |Ω t | p(t) -P (t) L 2 (Ωt) ∇w h (t) L 2 (Ωt) 4 + 1 |Ω t | p(t) L 2 (Ωt) ∇(w h (t) -w(t)) L 2 (Ωt) 4 .
This inequality together with (2.13) and (4.2) yields to

|λ -λ h | ≤ Ch| ln h|( u(t) H 3 (Ωt) 2 + p(t) H 2 (Ωt) ).
Using this estimate, (4.15) becomes

dU dt h Y (t) - du dt Y (t) H 1 (Ωt) 2 + dP dt h Y (t) - dp dt Y (t) L 2 (Ωt) ≤ Ch| ln h| u(t) H 3 (Ωt) 2 + du dt Y (t) H 2 (Ωt) 2 + p(t) H 2 (Ωt) + dp dt Y (t) H 1 (Ωt)
. (4.16)

Therefore, the estimate (4.5) is a direct consequence of (4.16). In fact, we have that

dU dt h Y (t) - du dt h Y (t) H 1 (Ωt) 2 + dP dt h Y (t) - dp dt h Y (t) L 2 (Ωt) ≤ dU dt h Y (t) - du dt Y (t) H 1 (Ωt) 2 + dp dt h Y (t) - dp dt Y (t) L 2 (Ωt) + ((w h (t) -w(t)) • ∇) u(t) H 1 (Ωt) 2 + (w h (t) -w(t)) • ∇p(t) L 2 (Ωt) ≤ dU dt h Y (t) - du dt Y (t) H 1 (Ωt) 2 + dp dt h Y (t) - dp dt Y (t) L 2 (Ωt) + C w h (t) -w(t) W 1,∞ (Ωt) 2 u(t) H 2 (Ωt) 2 + p(t) H 1 (Ωt) ,
and we conclude by combining (2.13) and (4.16).

Proof of rst main result

In this section we prove Theorem 2.1 by using the results of the previous section. Since (u, p) is solution of (2.4), then we have

                           d dt Ω t u(t) • v h • X -1 h,t dx + ν Ω t ∇u(t) : ∇ v h • X -1 h,t dx - Ωt div (w h (t) ⊗ u(t)) • v h • X -1 h,t dx - Ωt p(t) div v h • X -1 h,t dx = Ω t f (t) • v h • X -1 h,t dx ∀v h ∈ (W h,0 ) 2 , Ω t q h • X -1 h,t div u(t)dx = 0 ∀q h ∈ M h,0 , u(0) = u 0 in Ω 0 .
(5.1) Subtracting (2.11) from (5.1) and introducing the projections U(t) ∈ (W h,t ) 2 , P (t) ∈ M 0 h,t of the exact solutions u(t), p(t) dened in Proposition 4.1, we obtain

                                 d dt Ωt (u(t) -u h (t)) • v h • X -1 h,t dx + ν Ωt ∇ (U(t) -u h (t)) : ∇ v h • X -1 h,t dx - Ωt div w h (t) ⊗ (u(t) -u h (t)) • v h • X -1 h,t dx - Ωt (P (t) -p h (t)) div v h • X -1 h,t dx = Ωt f (t) • v h • X -1 h,t dx -I h,t f (t) • v h • X -1 h,t ∀v h ∈ (W h,0 ) 2 , Ωt q h • X -1 h,t div (U(t) -u h (t)) dx = 0 ∀q h ∈ M h,0 , u(0) -u h (0) = u 0 -u h,0 in Ω 0 .
For the time derivative of the rst integral, we apply formula (3.1) to obtain

                             Ωt du dt h Y (t) - du h dt h Y (t) • v h • X -1 h,t dx + ν Ωt ∇(U(t) -u h (t)) • ∇ v h • X -1 h,t dx - Ωt (w h (t) • ∇) (u(t) -u h (t)) • v h • X -1 h,t dx - Ωt (P (t) -p h (t))div v h • X -1 h,t dx = Ωt f (t) • v h • X -1 h,t dx -I h,t f (t) • v h • X -1 h,t ∀v h ∈ (W h,0 ) 2 , Ωt q h • X -1 h,t div (U(t) -u h (t)) dx = 0 ∀q h ∈ M h,0 , u(0) -u h (0) = u 0 -u h,0
in Ω 0 .

(5.2)

Using Proposition 3.1, we can choose in the above system the test functions (v h , q h ) such that

v h • X -1 h,t = U(t) -u h (t) ∈ (W h,t ) 2 , q h • X -1 h,t = P (t) -p h (t) ∈ M h,t .
Then it follows that

Ωt du dt h Y (t) - du h dt h Y (t) • (U(t) -u h (t))dx + ν Ωt |∇(U(t) -u h (t))| 2 dx - Ωt (w h (t) • ∇) (u(t) -u h (t)) • (U(t) -u h (t)) dx = Ωt f (t) • (U(t) -u h (t)) dx -I h,t (f (t) • (U(t) -u h (t))) (5.3)
On the other hand, due to the Reynolds formula, it can be checked that

1 2 d dt U(t) -u h (t) 2 L 2 (Ωt) 2 = Ωt dU dt h Y (t) - du h dt h Y (t) • (U(t) -u h (t)) dx - Ωt (w h (t) • ∇)(U(t) -u h (t)) • (U(t) -u h (t)) dx.
Combining this identity with (5.3), we obtain that:

1 2 d dt U(t) -u h (t) 2 L 2 (Ωt) 2 + ν ∇(U(t) -u h (t)) 2 L 2 (Ωt) 4 = 3 i=1 T i , (5.4)
where the terms T 1 , T 2 and T 3 are dened as follows:

T 1 = - Ωt du dt h Y (t) - dU dt h Y (t) • (U(t) -u h (t))dx, T 2 = Ωt (w h (t) • ∇) (u(t) -U(t)) • (U(t) -u h (t)) dx, T 3 = Ωt f (t) • (U(t) -u h (t)) dx -I h,t (f (t) • (U(t) -u h (t))) .
Now, let us estimate separately each term. Due to the Cauchy-Schwarz inequality and Theorem 4.3 we get that the rst term is bounded as follows:

T 1 ≤ Ch| ln h| u(t) H 3 (Ω t ) 2 + du dt Y (t) H 2 (Ωt) 2 + p(t) H 2 (Ω t ) + dp dt Y (t) H 1 (Ωt) • U(t) -u h (t) L 2 (Ωt) 2 .
The next term can be bounded using the Cauchy-Schwarz inequality, the estimates (2.13), (4.2) and we obtain that

T 2 ≤ Ch 2 w(t) W 2,∞ (Ωt) 2 u(t) H 3 (Ωt) 2 + p(t) H 2 (Ωt) U(t) -u h (t) L 2 (Ωt) 2 .
Now, let us estimate T 3 . Using the fact that

Ω t = K∈T h,t
K we can write

T 3 = Ωt f (t) • (U(t) -u h (t)) dx -I h,t (f (t) • (U(t) -u h (t))) = K∈T h,t E K (f (t) • (U(t) -u h (t))) ,
where E K represents the quadrature error on triangle K. To estimate this term, we apply Theorem 4.1.5 from Ciarlet [10, p. 195] and we obtain that for any q > 2,

T 3 ≤ Ch 2 K∈T h,t |K| 1/2-1/q f (t) W 2,q (K) 2 U(t) -u h (t) H 1 (K) 2 .
Combining the above inequality and the Hölder inequality with 1 2 + 1 p + 1 q = 1 , it follows that

T 3 ≤ Ch 2   K∈T h,t |K| 1 2 -1 q p   1/p   K∈T h,t f (t) q W 2,q (K) 2   1/q   K∈T h,t U(t) -u h (t) 2 H 1 (K) 2   1/2 ≤ Ch 2 f (t) W 2,q (Ω t ) 2 U(t) -u h (t) H 1 (Ω t ) 2 .
By using all previous bounds and the Poincaré inequality, (5.4) becomes

1 2 d dt U(t) -u h (t) 2 L 2 (Ωt) 2 + ν ∇(U(t) -u h (t)) 2 L 2 (Ωt) 4 ≤ Ch| ln h| u(t) H 3 (Ωt) 2 + du dt Y (t) H 2 (Ω t ) 2 + p(t) H 2 (Ωt) + dp dt Y (t) H 1 (Ω t ) + f (t) W 2,q (Ωt) 2 ∇(U(t) -u h (t)) L 2 (Ωt) 4 .
Now, integrating the above inequality, from 0 to t, we get

1 2 U(t) -u h (t) 2 L 2 (Ωt) 2 + ν t 0 ∇(U(s) -u h (s)) 2 L 2 (Ωs) 4 ds ≤ 1 2 U(0) -u h (0) 2 L 2 (Ω 0 ) 2 + Ch| ln h| t 0 u(s) H 3 (Ωs) 2 + du dt Y (s) H 2 (Ωs) 2 + p(s) H 2 (Ω s ) + dp dt Y (s) H 1 (Ωs) + f (s) W 2,q (Ω s ) 2 ∇(U(s) -u h (s)) L 2 (Ω s ) 4 ds,
then, due to the inequality ab ≤ 5 2ν a 2 + ν 10 b 2 ∀a, b ∈ R, we obtain that for all t ∈ (0, T ),

1 2 U(t) -u h (t) 2 L 2 (Ωt) 2 + ν 2 t 0 ∇(U(s) -u h (s)) 2 L 2 (Ωs) 4 ds ≤ 1 2 U(0) -u h (0) 2 L 2 (Ω 0 ) 2 + Ch 2 | ln h| 2 t 0 u(s) 2 H 3 (Ωs) 2 + du dt Y (s) 2 H 2 (Ωs) 2 + p(s) 2 H 2 (Ωs) + dp dt Y (s) 2 H 1 (Ωs) + f (s) 2 W 2,q (Ωs) 2 ds.
Hence,

1 2 U -u h 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + ν 2 ∇(U -u h ) 2 L 2 (0,T ;L 2 (Ωt) 4 ) ≤ 1 2 U(0) -u h (0) 2 L 2 (Ω 0 ) 2 + Ch 2 | ln h| 2 u 2 L 2 (0,T ;H 3 (Ωt) 2 ) + du dt Y 2 L 2 (0,T ;H 2 (Ω t ) 2 ) + p 2 L 2 (0,T ;H 2 (Ωt)) + dp dt Y 2 L 2 (0,T ;H 1 (Ω t ))
+ f 2 L 2 (0,T ;W 2,q (Ωt) 2 ) . (5.5)

In order to obtain the estimation (2.16), let us rst observe that

1 4 u -u h 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + ν 4 ∇(u -u h ) 2 L 2 (0,T ;L 2 (Ωt) 4 ) ≤ 1 2 u -U 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + ν 2 ∇(u -U) 2 L 2 (0,T ;L 2 (Ωt) 4 ) + 1 2 U -u h 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + ν 2 ∇(U -u h ) 2 L 2 (0,T ;L 2 (Ωt) 4 ) . (5.6)
On the other hand, since (4.2) holds true for each t ∈ (0, T ), we get that

1 2 u -U 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + ν 2 ∇(u -U) 2 L 2 (0,T ;L 2 (Ωt) 4 ) ≤ Ch 4 u 2 L ∞ (0,T ;H 3 (Ωt) 2 ) + p 2 L ∞ (0,T ;H 2 (Ωt))
(5.7) and

1 2 U(0) -u h (0) 2 L 2 (Ω 0 ) 2 ≤ u(0) -U(0) 2 L 2 (Ω 0 ) 2 + u(0) -u h (0) 2 L 2 (Ω 0 ) 2 ≤ Ch 4 u(0) 2 H 3 (Ω 0 ) 2 + p(0) 2 H 2 (Ω 0 ) + u(0) -u h (0) 2 L 2 (Ω 0 ) 2 . (5.8)
By using (5.5)(5.8), we get the result stated in Theorem 2.1.

6 Proof of second main result

In this section, we will analyze the full discretization of the problem (2.5) given in (2.11). We will prove that the numerical solution converges to the exact solution of the problem, when the discretization parameters ∆t and h go to zero, if a compatibility condition between ∆t and h is fullled.

6.1 Proof of Theorem 2.2

We remark that the approximation error u(t n+1 ) -U(t n+1 ) is well known, and is given in the estimate (4.2). For this reason, we will study the following error:

e n+1 h = U(t n+1 ) -u n+1 h ∀n = 0, . . . , N -1. (6.1)
Since (u, p) is solution of (2.4), we have that

                           d dt Ωt u(t) • v h • X -1 h,t dx + ν Ωt ∇u(t) : ∇ v h • X -1 h,t dx - Ω t p(t)div v h • X -1 h,t dx - Ω t div (w h (t) ⊗ u(t)) • v h • X -1 h,t dx = Ω t f (t) • v h • X -1 h,t dx ∀v h ∈ (W h,0 ) 2 , Ωt q h • X -1 h,t div u(t) dx = 0 ∀q h ∈ M h,0 , u(0) = u 0 in Ω 0 . (6.2)
Then, integrating the rst equation of the above system from t n to t n+1 , we get

Ω t n+1 u(t n+1 ) • v h • X -1 h,t n+1 dx - Ω tn u(t n ) • v h • X -1 h,tn dx + ν t n+1 tn Ωt ∇u(t) : ∇ v h • X -1 h,t dx dt - t n+1 tn Ωt p(t)div v h • X -1 h,t dx dt - t n+1 tn Ωt div (w h (t) ⊗ u(t)) • v h • X -1 h,t dx dt = t n+1 t n Ω t f (t) • v h • X -1 h,t dx dt ∀v h ∈ (W h,0 ) 2 .
The previous identity could be rewritten similarly to the numerical equations as follows:

Ωt n+1 u(t n+1 ) • v h • X -1 h,t n+1 dx - Ωt n u(t n ) • v h • X -1 h,tn dx + ∆t ν Ωt n+1 ∇u(t n+1 ) : ∇ v h • X -1 h,t n+1 dx -∆t Ωt n+1 p(t n+1 )div v h • X -1 h,t n+1 dx -∆t Ωt n+1 div (w h (t n+1 ) ⊗ u(t n+1 )) • v h • X -1 h,t n+1 dx = ∆t Ωt n+1 f (t n+1 ) • v h • X -1 h,t n+1 dx + 4 i=1 Q i ∀v h ∈ (W h,0 ) 2 , (6.3)
where Q i (i = 1, . . . , 4) are the dierences between the time integrals and the numerical approximations given by the right point integration formula. That is,

Q 1 = ν ∆t Ωt n+1 ∇u(t n+1 ) : ∇ v h • X -1 h,t n+1 dx -ν t n+1 tn Ωt ∇u(t) : ∇ v h • X -1
h,t dx dt, (6.4)

Q 2 = -∆t Ωt n+1 div (w h (t n+1 ) ⊗ u(t n+1 )) • v h • X -1 h,t n+1 dx + t n+1 tn Ωt div (w h (t) ⊗ u(t)) • v h • X -1
h,t dx dt, (6.5)

Q 3 = -∆t Ω t n+1 p(t n+1 )div v h • X -1 h,t n+1 dx + t n+1 t n Ω t p(t)div v h • X -1 h,t dx dt, (6.6
)

Q 4 = -∆t Ωt n+1 f (t n+1 ) • v h • X -1 h,t n+1 dx + t n+1 tn Ωt f (t) • v h • X -1 h,t dx dt. (6.7)
Using the projections of u(t n+1 ) and p(t n+1 ), denoted by

U(t n+1 ) ∈ (W h,t n+1 ) 2 and P (t n+1 ) ∈ M 0 h,t n+1 ,
and dened in (4.3), the problem (6.2) can be written as follows:

                                                 Ω t n+1 u(t n+1 ) • v h • X -1 h,t n+1 dx - Ωt n u(t n ) • v h • X -1 h,tn dx + ∆t ν Ωt n+1 ∇U(t n+1 ) : ∇ v h • X -1 h,t n+1 dx -∆t Ωt n+1 div (w h (t n+1 ) ⊗ u(t n+1 )) • v h • X -1 h,t n+1 dx -∆t Ωt n+1 P (t n+1 )div v h • X -1 h,t n+1 dx = ∆t Ωt n+1 f (t n+1 ) • v h • X -1 h,t n+1 dx + 4 i=1 Q i ∀v h ∈ (W h,0 ) 2 , Ωt n+1 q h • X -1 h,t n+1 div U(t n+1 ) dx = 0 ∀q h ∈ M h,0 , u(0) = u 0 in Ω 0 . (6.8)
The preceding system allows us to compare directly the numerical solution with the exact one: by subtracting (6.8) and (2.17) we get

                                                               Ωt n+1 u(t n+1 ) -u n+1 h • v h • X -1 h,t n+1 dx - Ωt n (u(t n ) -u n h ) • v h • X -1 h,tn dx +∆t ν Ωt n+1 ∇ U(t n+1 ) -u n+1 h : ∇ v h • X -1 h,t n+1 dx -∆t Ωt n+1 div (w h (t n+1 ) ⊗ u(t n+1 )) • v h • X -1 h,t n+1 dx +∆t Ωt n+1 div w * h,n,n+1 ⊗ u n+1 h • v h • X -1 h,t n+1 dx -∆t Ωt n+1 P (t n+1 ) -p n+1 h div v h • X -1 h,t n+1 dx = 4 i=1 Q i + ∆t Ωt n+1 f (t n+1 ) • v h • X -1 h,t n+1 dx -∆t I h,t n+1 f (t n+1 ) • (v h • X -1 h,t n+1 ) ∀v h ∈ (W h,0 ) 2 , Ωt n+1 q h • X -1 h,t n+1 div U(t n+1 ) -u n+1 h dx = 0 ∀q h ∈ M h,0 , u(0) -u 0 h = u 0 -u h,0 in Ω 0 . (6.9)
We note that in the previous problem there are two convective terms, with the velocities w h and w * h,n,n+1 . In order to compare these two velocities, we use the denition of w * h,n,n+1 , and therefore we get

w h (x, t n+1 ) = w * h,n,n+1 (x) + 1 ∆t t n+1 t n (s -t n ) ∂ 2 X h ∂s 2 X -1 h,t n+1 (x), s ds. (6.10)
Gathering this identity and (6.9), then by using the notation (6.1) it follows the following system:

                                                                                 Ωt n+1 (u(t n+1 ) -U(t n+1 )) • v h • X -1 h,t n+1 dx - Ωt n (u(t n ) -U(t n )) • v h • X -1 h,tn dx + Ω t n+1 e n+1 h • v h • X -1 h,t n+1 dx - Ω tn e n h • v h • X -1 h,tn dx +∆t ν Ω t n+1 ∇e n+1 h : ∇ v h • X -1 h,t n+1 dx -∆t Ω t n+1 div w * h,n,n+1 ⊗ e n+1 h • v h • X -1 h,t n+1 dx - Ωt n+1 div t n+1 tn (s -t n ) ∂ 2 X h ∂s 2 X -1 h,t n+1 (x), s ds ⊗ u(t n+1 ) • v h • X -1 h,t n+1 dx -∆t Ωt n+1 P (t n+1 ) -p n+1 h div v h • X -1 h,t n+1 dx = 4 i=1 Q i + ∆t Ωt n+1 f (t n+1 ) • v h • X -1 h,t n+1 dx -∆t I h,t n+1 f (t n+1 ) • (v h • X -1 h,t n+1 ) ∀v h ∈ (W h,0 ) 2 , Ω t n+1 q h • X -1 h,t n+1 div e n+1 h dx = 0 ∀q h ∈ M h,0 , u(0) -u 0 h = u 0 -u h,0 in Ω 0 . (6.11)
In the above system, we choose the following test functions:

v h = e n+1 h • X h,t n+1 ∈ (W h,0 ) 2 , q h = P (t n+1 ) -p n+1 h • X h,t n+1 ∈ M h,0
and we get

e n+1 h 2 L 2 (Ωt n+1 ) 2 + ν ∆t ∇e n+1 h 2 L 2 (Ωt n+1 ) 4 = 9 j=1 R j , (6.12)
where the right hand side is given by

R 1 = Ωt n e n h • e n+1 h • X h,t n+1 • X -1 h,tn dx + ∆t Ωt n+1 div w * h,n,n+1 ⊗ e n+1 h • e n+1 h dx, (6.13)
R 2 = Ωt n (u(t n ) -U(t n )) • e n+1 h • X h,t n+1 • X -1 h,tn dx - Ω t n+1 (u(t n+1 ) -U(t n+1 )) • e n+1 h dx, (6.14)
R 3 = -∆t Ωt n+1 div w * h,n,n+1 ⊗ (u(t n+1 ) -U(t n+1 )) • e n+1 h dx, (6.15)
R 4 = Ωt n+1 div t n+1 tn (s -t n ) ∂ 2 X h ∂s 2 X -1 h,t n+1 (x), s ds ⊗ u(t n+1 ) • e n+1 h dx, (6.16
)

R 5 = ν ∆t Ωt n+1 ∇u(t n+1 ) : ∇e n+1 h dx -ν t n+1 tn Ωt ∇u(t) : ∇ e n+1 h • X h,t n+1 • X -1 h,t dx dt, (6.17)
R 6 = -∆t Ωt n+1 div (w h (t n+1 ) ⊗ u(t n+1 )) • e n+1 h dx + t n+1 tn Ωt div (w h (t) ⊗ u(t)) • e n+1 h • X h,t n+1 • X -1 h,t dx dt, (6.18)
R 7 = -∆t Ωt n+1 p(t n+1 )div e n+1 h dx + t n+1 tn Ωt p(t)div e n+1 h • X h,t n+1 • X -1 h,t dx dt, (6.19)
R 8 = -∆t Ωt n+1 f (t n+1 ) • e n+1 h dx + t n+1 tn Ωt f (t) • e n+1 h • X h,t n+1 • X -1 h,t dx dt, (6.20)
R 9 = ∆t Ω t n+1 f (t n+1 ) • e n+1 h dx -∆t I h,t n+1 f (t n+1 ) • e n+1 h . (6.21)
The estimates of the terms R i (i = 1, . . . , 9) are very technical, and we prefer to postpone their proof to Subsection 6.2. For the sake of completeness, in the sequel we present the results obtained, nevertheless, the precise results are stated in Lemmas 6.16.4 below. We get that

R 1 ≤ 1 2 e n h 2 L 2 (Ω tn ) 2 + 1 2 e n+1 h 2 L 2 (Ω t n+1 ) 2 + 1 2 ∆t γ e n+1 h 2 L 2 (Ω t n+1 ) 2 , (6.22)
where

γ = max n=0,...,N -1 sup t∈(tn,t n+1 ) div w h (t) L ∞ (Ω t ) sup t∈(tn,t n+1 ) J X h,t L ∞ (Ω 0 ) J X -1 h,t n+1 L ∞ (Ω t n+1 ) + div w * h,n,n+1 L ∞ (Ω t n+1 ) . Furthermore, R 2 ≤ C 1 ∆t u -U 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + 1 18 ν∆t ∇e n+1 h 2 L 2 (Ωt n+1 ) 4 , (6.23) R 3 ≤ C∆t u -U 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + 1 18 ν∆t ∇e n+1 h 2 L 2 (Ωt n+1 ) 4 , (6.24) R 4 ≤ C∆t 3 sup s∈(tn,t n+1 ) ∂ 2 X h ∂s 2 (s) 2 L ∞ (Ω 0 ) u 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + 1 18 ν∆t ∇e n+1 h 2 L 2 (Ωt n+1 ) 4 .(6.25)
In addition,

R 5 ≤ C∆t 2 t n+1 tn   u(t) 2 H 1 (Ωt) 2 + du dt h Y (t) 2 H 1 (Ωt) 2   dt + 1 18 ν∆t ∇e n+1 h 2 L 2 (Ωt n+1 ) 4 , (6.26) R 6 ≤ C∆t 2 t n+1 t n   u(t) 2 H 1 (Ωt) 2 + du dt h Y (t) 2 L 2 (Ωt) 2   dt + 1 9 ν∆t ∇e n+1 h 2 L 2 (Ωt n+1 ) 4 , (6.27) R 7 ≤ C∆t 2 t n+1 tn   p(t) 2 L 2 (Ωt) + dp dt h Y (t) 2 L 2 (Ωt)   dt + 1 18 ν∆t ∇e n+1 h 2 L 2 (Ωt n+1 ) 4 , (6.28)
R 8 ≤ C∆t 2 t n+1 tn   f (t) 2 L 2 (Ωt) 2 + df dt h Y (t) 2 L 2 (Ωt) 2   dt + 1 18 ν∆t ∇e n+1 h 2 L 2 (Ωt n+1 ) 4 (6.29)
and

R 9 ≤ C∆t h 4 f (t n+1 ) 2 W 2,q (Ωt n+1 ) 2 + 1 18 ν∆t ∇e n+1 h 2 L 2 (Ωt n+1 ) 4 .
(6.30)

By using these estimates of R i (i = 1, . . . , 9) in (6.12), we obtain

e n+1 h 2 L 2 (Ωt n+1 ) 2 + ν ∆t ∇e n+1 h 2 L 2 (Ωt n+1 ) 4 ≤ ∆t γ e n+1 h 2 L 2 (Ωt n+1 ) 2 + e n h 2 L 2 (Ωt n ) 2 + C 1 ∆t + ∆t u -U 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + C∆t h 4 f (t n+1 ) 2 W 2,q (Ωt n+1 ) 2 + C∆t 3 sup s∈(tn,t n+1 ) ∂ 2 X h ∂s 2 (s) 2 L ∞ (Ω 0 ) u 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + C∆t 2 t n+1 tn u(t) 2 H 1 (Ωt) 2 + du dt h Y (t) 2 H 1 (Ωt) 2 + p(t) 2 L 2 (Ωt) + dp dt h Y (t) 2 L 2 (Ω t ) + f (t) 2 L 2 (Ωt) 2 + df dt h Y (t) 2 L 2 (Ω t ) 2
dt.

In order to obtain the global error, we sum over n, that is,

e n+1 h 2 L 2 (Ωt n+1 ) 2 + ν ∆t n+1 i=1 ∇e i h 2 L 2 (Ωt i ) 4 ≤ e 0 h 2 L 2 (Ω 0 ) 2 + ∆t γ n+1 i=1 e i h 2 L 2 (Ωt i ) 2 + Cn 1 ∆t + ∆t u -U 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + C∆t h 4 n+1 i=1 f (t i ) 2 W 2,q (Ωt i ) 2 + Cn∆t 3 sup s∈(0,T ) ∂ 2 X h ∂s 2 (s) 2 L ∞ (Ω 0 ) u 2 L ∞ (0,T ;L 2 (Ω t ) 2 ) + C∆t 2 t n+1 0 u(t) 2 H 1 (Ωt) 2 + du dt h Y (t) 2 H 1 (Ωt) 2 + p(t) 2 L 2 (Ω t ) + dp dt h Y (t) 2 L 2 (Ωt) + f (t) 2 L 2 (Ω t ) 2 + df dt h Y (t) 2 L 2 (Ωt) 2 dt.
By applying the discrete Gronwall lemma, we get

e n+1 h 2 L 2 (Ωt n+1 ) 2 + ν ∆t n+1 i=1 ∇e i h 2 L 2 (Ωt i ) 4 ≤ C 1 e 0 h 2 L 2 (Ω 0 ) 2 + CC 1 T 1 ∆t 2 + 1 u -U 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + CC 1 ∆t h 4 n+1 i=1 f (t i ) 2 W 2,q (Ωt i ) 2 + CC 1 T ∆t 2 sup s∈(0,T ) ∂ 2 X h ∂s 2 (s) 2 L ∞ (Ω 0 ) u 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + CC 1 ∆t 2 t n+1 0 u(t) 2 H 1 (Ωt) 2 + du dt h Y (t) 2 H 1 (Ωt) 2 + p(t) 2 L 2 (Ω t ) + dp dt h Y (t) 2 L 2 (Ωt) + f (t) 2 L 2 (Ω t ) 2 + df dt h Y (t) 2 L 2 (Ωt) 2
dt, (6.31) where the constant C 1 is given by

C 1 = exp t n+1 γ 1 -γ∆t .
In the previous estimate, we will introduce the continuous ALE derivatives using the identities (4.7), (4.8) and

df dt h Y (t) = df dt Y (t) + ((w h (t) -w(t)) • ∇) f (t).
Therefore, the estimate (6.31) becomes

e n+1 h 2 L 2 (Ωt n+1 ) 2 + ν ∆t n+1 i=1 ∇e i h 2 L 2 (Ωt i ) 4 ≤ C e 0 h 2 L 2 (Ω 0 ) 2 + C 1 ∆t 2 + 1 u -U 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + C∆t h 4 n+1 i=1 f (t i ) 2 W 2,q (Ωt i ) 2 + C∆t 2 sup s∈(0,T ) ∂ 2 X h ∂s 2 (s) 2 L ∞ (Ω 0 ) u 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + C∆t 2 t n+1 0 u(t) 2 H 2 (Ω t ) 2 + du dt Y (t) 2 H 1 (Ωt) 2 + p(t) 2 H 1 (Ω t ) + dp dt Y (t) 2 L 2 (Ωt) + f (t) 2 H 1 (Ω t ) 2 + df dt Y (t) 2 L 2 (Ωt) 2
dt. (6.32) This inequality gives us the numerical error U(t n+1 )u n+1 h . In order to obtain the complete error, we observe that

u(t n+1 ) -u n+1 h 2 L 2 (Ωt n+1 ) 2 + ν ∆t n+1 i=1 ∇ u(t i ) -u i h 2 L 2 (Ωt i ) 4 ≤ 1 2 u(t n+1 ) -U(t n+1 ) 2 L 2 (Ω t n+1 ) 2 + 1 2 ν ∆t n+1 i=1 ∇ (u(t i ) -U(t i )) 2 L 2 (Ω t i ) 4 + 1 2 e n+1 h 2 L 2 (Ωt n+1 ) 2 + 1 2 ν ∆t n+1 i=1 ∇e i h 2 L 2 (Ωt i ) 4 .
Combining the previous inequalities and using (4.2), we conclude the proof of the second main result of this paper.

Some additional estimates

In this subsection, we derive estimates on R i (i = 1, . . . , 9) which have been used in the proof of Theorem 2.2.

Lemma 6.1. Suppose that the assumptions of Theorem 2.2 hold true. Then, the terms R 1 and R 2 dened in (6.13) and (6.14) satisfy (6.22), respectively (6.23).

Proof. By using the Cauchy-Schwarz inequality, we have that

R 1 ≤ 1 2 e n h 2 L 2 (Ωt n ) 2 + 1 2 e n+1 h • X h,t n+1 • X -1 h,tn 2
L 2 (Ωt n ) 2 + ∆t Ω t n+1 div w * h,n,n+1 ⊗ e n+1 h • e n+1 h dx,
then, integrating twice by parts, we obtain

R 1 ≤ 1 2 e n h 2 L 2 (Ω tn ) 2 + 1 2 e n+1 h • X h,t n+1 • X -1 h,t n 2 L 2 (Ω tn ) 2 + 1 2 ∆t Ωt n+1 |e n+1 h | 2 div w * h,n,n+1 dx. (6.33)
In order to transform the second term in the right hand side, we use the Reynolds formula:

d dt Ωt e n+1 h • X h,t n+1 • X -1 h,t 2 dx = Ωt e n+1 h • X h,t n+1 • X -1 h,t 2 div w h (t)dx.
Therefore, integrating from t n to t n+1 , we get

e n+1 h 2 L 2 (Ωt n+1 ) 2 -e n+1 h • X h,t n+1 • X -1 h,tn 2
L 2 (Ωt n ) 2 = t n+1 tn Ωt e n+1 h • X h,t n+1 • X -1 h,t 2
div w h (t) dx dt.

By combining the above equation with (6.33), we get

R 1 ≤ 1 2 e n h 2 L 2 (Ωt n ) 2 + 1 2 e n+1 h 2 L 2 (Ωt n+1 ) 2 + 1 2 ∆t Ωt n+1 |e n+1 h | 2 div w * h,n,n+1 dx - 1 2 t n+1 t n Ω t e n+1 h • X h,t n+1 • X -1 h,t 2 div w h (t) dx dt.
Hence, we get that

R 1 ≤ 1 2 e n h 2 L 2 (Ωt n ) 2 + 1 2 e n+1 h 2 L 2 (Ωt n+1 ) 2 + 1 2 ∆t div w * h,n,n+1 L ∞ (Ωt n+1 ) e n+1 h 2 L 2 (Ωt n+1 ) 2 + 1 2 sup t∈(tn,t n+1 ) div w h (t) L ∞ (Ω t ) t n+1 t n e n+1 h • X h,t n+1 • X -1 h,t 2 L 2 (Ωt) 2 dt. (6 .34)
In order to bound the last integral, let us remark that, due to the change of variable y = X h,t n+1 X -1 h,t (x) ,

we have that

e n+1 h • X h,t n+1 • X -1 h,t 2
L 2 (Ωt) 2 ≤ J X h,t L ∞ (Ω 0 ) J X -1 h,t n+1 L ∞ (Ωt n+1 ) e n+1 h 2 L 2 (Ωt n+1 ) 2 . (6.35) Let us observe that J X h,t L ∞ (Ω 0 ) = det J X h,t L ∞ (Ω 0 ) ≤ C J Xt 2 L ∞ (Ω 0 ) 4 + Ch| ln h| X t W 2,∞ (Ω 0 ) 2 .
Thus, there exists C 1 depending on X and h 0 > 0 such that

J X h,t L ∞ (Ω 0 ) ≤ C 1 ∀t ∈ [0, T ], ∀h ∈ (0, h 0 ). (6.36)
We can prove in a similar way that there exists C 2 depending on X and h 0 > 0 such that

J X -1 h,t L ∞ (Ωt) ≤ C 2 ∀t ∈ [0, T ], ∀h ∈ (0, h 0 ). (6.37)
From (6.35), (6.36) and (6.37), we obtain

e n+1 h • X h,t n+1 • X -1 h,t 2
L 2 (Ω t ) 2 ≤ C 1 C 2 e n+1 h 2 L 2 (Ω t n+1 ) 2 . (6.38)
Combining the above inequality with (6.34) we get (6.22).

Let us estimate the term R 2 . The Cauchy-Schwarz inequality together with (6.38) yields to

R 2 ≤ C u -U L ∞ (0,T ;L 2 (Ωt) 2 ) e n+1 h L 2 (Ωt n+1 ) 2 .
To conclude, it is enough to use the Poincaré inequality and that

ab ≤ 2 9 a 2 + 1 18 b 2 ∀a, b ∈ R.
(6.39) Lemma 6.2. Suppose that the assumptions of Theorem 2.2 hold true. Then, the terms R 3 and R 4 dened in (6.15) and (6.16) satisfy (6.24), respectively (6.25).

Proof. To estimate R 3 , rst we integrate by parts:

R 3 = -∆t Ωt n+1 w * h,n,n+1 • ∇ e n+1 h • (u(t n+1 ) -U(t n+1 )) dx.
Then, by the Cauchy-Schwarz inequality and (6.39), we obtain

R 3 ≤ 4 ν ∆t w * h,n,n+1 2
L ∞ (Ωt n+1 ) 2 u -U 2 L ∞ (0,T ;L 2 (Ωt) 2 ) + 1 18 ν∆t ∇e n+1 h 2 L 2 (Ωt n+1 ) 4 ,
which implies the estimate (6.24).

Let us estimate the term R 4 . First, we integrate by parts and use the Einstein notation:

R 4 = - Ωt n+1 t n+1 tn (s -t n ) ∂ 2 X h ∂s 2 X -1 h,t n+1 (x), s ds • ∇ e n+1 h • u(t n+1 ) dx = - t n+1 tn (s -t n ) Ωt n+1 ∂ 2 X h ∂s 2 X -1 h,t n+1 (x), s j ∂(e n+1 h ) i ∂x j u i (t n+1 ) dx ds.
In order to write the integral in the domain Ω 0 , we use the change of variable X -1 h,t n+1 (x) = y ∈ Ω 0 , then it follows that

R 4 = - t n+1 tn (s -t n ) Ω 0 ∂ 2 X h ∂s 2 (y, s) j ∂ e n+1 h • X h,t n+1 i ∂y k • ∂ X -1 h,t n+1 k ∂x j u(t n+1 ) • X h,t n+1 i J X h,t n+1 dy ds.
Applying the Cauchy-Schwarz inequality, we have

R 4 ≤ t n+1 tn |s -t n |   Ω 0 ∂ e n+1 h • X h,t n+1 i ∂y k ∂y k ∂x j 2 J X h,t n+1 dy   1/2 • Ω 0 ∂ 2 X h ∂s 2 (y, s) 2 j u(t n+1 ) • X h,t n+1 2 i J X h,t n+1 dy 1/2 ds,
and therefore,

R 4 ≤ t n+1 tn |s -t n | ∇e n+1 h L 2 (Ω t n+1 ) 4 ∂ 2 X h ∂s 2 (s) 2 1/2 L ∞ (Ω 0 ) 2 u(t n+1 ) L 2 (Ω t n+1 ) 2 ds.
By using the Cauchy-Schwarz inequality in time, it follows that

R 4 ≤ ∆t 2 √ 3   sup s∈(tn,t n+1 ) ∂ 2 X h ∂s 2 (s) 2 L ∞ (Ω 0 ) 2   1/2 u L ∞ (0,T ;L 2 (Ωt) 2 ) ∇e n+1 h L 2 (Ωt n+1 ) 4 ,
which yields (6.25).

Lemma 6.3. Suppose that the assumptions of Theorem 2.2 hold true. Then, the terms R 5 to R 8 dened in (6.17)(6.20) satisfy (6.26)(6.29), respectively.

Proof. In order to simplify matters, let us start our proof by studying the terms Q 1 to Q 4 dened in (6.4)(6.7), which are basically the same as the terms R 5 to R 8 , but written for a general test function

v h ∈ (W h,0 ) 2 .
We will begin by rewriting the term Q 1 as follows: then, by the Poincaré inequality and (6.39), we get (6.27).

Q 1 = ν t n+1 tn Ωt n+1 ∇u(t n+1 ) : ∇(v h • X -1 h,t n+1 ) dx - Ωt ∇u(t) : ∇ v h • X -1 h,t dx dt = ν t n+1 tn t n+1 t d ds Ωs ∇u(s) : ∇(v h • X -1 h,
∇ e n+1 h • X h,t n+1 • X -1 h,t 2
L 2 (Ωt) 4 ≤ 2 4 J X h,t 2
L ∞ (Ω 0 ) 4 J X -1 h,t L ∞ (Ωt) • J X -1 h,t n+1 2 L ∞ (Ωt n+1 ) 4 J X h,
Estimates (6.28) and (6.29) can be obtained in a similar way, then we skip their derivation. Lemma 6.4. Suppose that the assumptions of Theorem 2.2 hold true. Then, the term R 9 dened in (6.21) satises (6.30).

Proof. First of all, we observe that this term is similar with T 3 , which has been estimated in the proof of Theorem 2.1. Hence, we are going to proceed similarly. We have that

R 9 = ∆t Ωt n+1 f (t n+1 ) • e n+1 h dx -∆t I h,t n+1 f (t n+1 ) • e n+1 h = ∆t K∈T h,t n+1 E K f (t n+1 ) • e n+1 h .
In order to obtain this error, we use Theorem 4.1.5 from Ciarlet [10, p.195], then for any q > 2,

R 9 ≤ C∆t h 2 K∈T h,t n+1
|K| 1/2-1/q f (t n+1 ) W 2,q (K) 2 e n+1 h H 1 (K) 2 .

Now, applying the Hölder inequality with 1 2 + 1 p + 1 q = 1 , we get

R 9 ≤ C∆t h 2 K |K| 1 2 -1 q p 1 p K f (t n+1 ) q W 2,q (K) 2 1 q K e n+1 h 2 H 1 (K) 2 1 2
≤ C∆t h 2 f (t n+1 ) W 2,q (Ωt n+1 ) 2 e n+1 h H 1 (Ωt n+1 ) 2 .

To conclude, we combine the above relations with the Poincaré inequality and with (6.39).

(2. 2 )

 2 Then we use the notation dv dt Y for the time derivative on the ALE frame which is dened as follows: for any function v : Q T → R regular enough and dened on the Eulerian frame, we set

(4. 15 )

 15 In order to estimate the term |λ -λ h |, let us remark that Ωt p(t) dx = Ωt P (t) dx = 0 ∀t ∈ (0, T ),

  2 and ∈ H 1 (Ω t ). In order toproject them onto (W h,t ) 2 × M 0 h,t , since Now, since p1 (t) ∈ H 1 (Ω t ) ∩ L 2 0 (Ω t ), we can consider the projection U 1 (t), P 1 (t) ∈ (W h,t ) 2 × M 0 h,t of

	dened by	dp dt Y (t) dp dt Y 0 (Ω t ), we need to introduce an auxiliary function p1 (t) ∈ L 2
		p1 (t) =	dp dt Y	(t) -λ,
	where λ = Let us note that the equation (4.9a) is also true if we change 1 dp dt Y (t) dx. |Ω t | Ω t	dp dt Y	(t) by p1 (t).
	du
	dt Y

  By using this zero mean value projection and Remark 1.3 from Girault and Raviart (see[18, p.117]), we have that

						dx. (4.12b)
	In the system (4.12a)(4.12b), we can change	dP dt	h Y	(t) by the corresponding zero mean value projection
	P1 (t) dened by
					P1 (t) =	dP dt	h Y	(t) -λ h ,
	where λ h =	1 |Ω t | Ωt	dP dt	h Y	(t) dx.

  s ) dx ds dt. ) + p(s)div (v h • X -1 h,s )div w h (s) -p(s)∇w h (s) : ∇(v h • X -1 h,s ) dx ds dt, (6.42) Q 4 = -• X h,t n+1 • X -1 h,s )div w h (s) -∇w h (s) + ∇w h (s) T ∇u(s) : ∇(e n+1 h • X h,t n+1 • X -1 h,s ) dx ds dt. + 3 ∇w h (s) L ∞ (Ω s ) 4 ∇u(s) L 2 (Ω s ) 4

	Therefore, we have that
	R 5 = ν	t n+1 tn	t	t n+1	Ωs	∇	du dt	h Y	(s) : ∇(e n+1 h	• X h,t n+1 • X -1 h,s )
							+ ∇u(s) : ∇(e n+1 h
	Applying the Cauchy-Schwarz inequality, we obtain
	R 5 ≤ ν	t n+1 tn	t	t n+1	∇	du dt	h Y	(s)	L 2 (Ωs) 4
											• ∇ e n+1 h	• X h,t n+1 • X -1 h,s	L 2 (Ωs) 4 ds dt. (6.44)
	By using the following inequality
	Due to Lemma 3.2, it follows that
	Q 1 = ν	t n+1 tn	t	t n+1	Ωs	∇	du dt
		t n+1 t n	t	t n+1	Ω s	div		dw h dt
		t n+1 t n	t n+1 h,s t n+1 t Ω s h dp dt Y (s)div (v h • X -1 t n+1 df
			tn		t		Ωs		dt

h Y (s) : ∇(v h • X -1 h,s ) + ∇u(s) : ∇(v h • X -1 h,s )div w h (s) -∇w h (s) + ∇w h (s) T ∇u(s) : ∇(v h • X -1 h,s

) dx ds dt. (6.40)

Similarly, we deduce that

Q 2 = -h Y (s) ⊗ u(s) + div (w h (s) ⊗ u(s))div w h (s) + div w h (s) ⊗ du dt h Y (s) • v h • X -1

h,s dx ds dt, (6.41)

Q 3 =h Y (s) + f (s)div w h (s) • v h • X -1 h,s dx ds dt.

(6.43)

In order to obtain the estimates (6.26)(6.29), let us choose in the terms Q i , for all i = 1, . . . , 4, the following test function:

v h = e n+1 h • X h,t n+1 ∈ (W h,0 ) 2 .

  t n+1 L ∞ (Ω 0 ) + 3 ∇w h (s) L ∞ (Ω s ) 4 ∇u(s) L 2 (Ω s ) 4 Ωt n+1 ) 4 ds dt. Ωs div w h (s) (w h (s) • ∇) u(s) • e n+1 h • X h,t n+1 • X -1 h,s dx ds dt. L ∞ (Ωs) 4 u(s) L 2 (Ω s ) 2 + ∇w h (s) L ∞ (Ωs) 4 w h (s) L ∞ (Ωs) 2 ∇u(s) L 2 (Ωs) 4 e n+1 h L 2 (Ωt n+1 ) 2 ds dt.Using the estimate (2.13), the hypothesis (2.18) and the Cauchy-Schwarz inequality, it follows that R 6 ≤ C∆t 3/2 L 2 (Ωt) 2 + ∇u(t) L 2 (Ωt) 4

	then, integrating by parts, it follows that
	R 6 =	t n+1 tn	t	t n+1		Ωs		dw h dt		h Y	(s) • ∇	e n+1 h	• X h,t n+1 • X -1 h,s • u(s) dx ds dt
			+	t n+1 tn	t	t n+1	Ωs	(w h (s) • ∇) e n+1 h	• X h,t n+1 • X -1 h,s •	du dt	h Y	(s) dx ds dt
			-	t n+1 t n	t	t n+1		Ω s	[div w h (s)] 2 u(s) • e n+1 h	• X h,t n+1 • X -1 h,s dx ds dt
							t n+1	t n+1
				-		tn			t
	Applying the Cauchy-Schwarz inequality, (6.35) and (6.45), we get
	R 6 ≤ C	t n+1 tn			t n+1 tn		dw h dt	h Y	(s)	L ∞ (Ωs) 2	u(s) L 2 (Ωs) 2
			+ w h (s) L ∞ (Ωs) 2	du dt	h Y	(s)	L 2 (Ωs) 2	∇e n+1 h	L 2 (Ωt n+1 ) 4 ds dt
								t n+1	t n+1
					+ C
								tn						∇e n+1 h	2 L 2 (Ωt n+1 ) 4 , (6.45)
	then (6.44) yields to
	R 5 ≤ C L 2 (By the Cauchy-Schwarz inequality and (2.13), the previous estimation becomes t n+1 tn t n+1 tn ∇ h du dt Y (s) L 2 (Ωs) 4 t n+1 tn u(t) L 2 (Ωt) 2 + h 2 1/2 du dt Y (t) L 2 (Ωt) 2 dt h L 2 (Ωt n+1 ) 4 ∇e n+1 • ∇e n+1 h + C∆t 3/2 t n+1 tn 1/2 u(t) 2 dt e n+1 h	L 2 (Ωt n+1 ) 2 ,
	R 5 ≤ C∆t 3/2	 	t n+1 tn	  ∇			du dt	h Y	(t)	L 2 (Ωt) 4	+ ∇u(t) L 2 (Ωt) 4	 2 	 dt 	1 2	∇e n+1 h	L 2 (Ωt n+1 ) 4 ,
	and therefore, due to (6.39), we get
	R 5 ≤ C∆t 2		t n+1 tn	  ∇	du dt	h Y	(t)	2 L 2 (Ωt) 4	+ ∇u(t) 2 L 2 (Ωt) 4	  dt +	1 18	ν∆t ∇e n+1 h	2 L 2 (Ωt n+1 ) 4 ,
	which completes the proof of the estimate (6.26).
	On the other hand, we have that
	R 6 = -	t n+1 tn		t	t n+1		Ωs	div			dw h dt
															h,s	dx ds dt,

h Y (s) ⊗ u(s) + div (w h (s) ⊗ u(s))div w h (s) +div w h (s) ⊗ du dt h Y (s) • e n+1 h • X h,t n+1 • X -1 tn ∇w h (s) 2

Acknowledgments

The rst author was partially supported by grants Fondecyt 1050332 and Fondap on Applied Mathematics. The second one was partially supported by grant Fondecyt 3070029 and fellowship of Center of Mathematical Modelling of University of Chile.