Esther V Uijtendaal
email: e.v.uijtendaal@umcutrecht.nl

Jeannette E F Zwart-Van Rijkom

Wouter W Van Solinge

C G Egberts

Drs E V Uijtendaal

Frequency of laboratory measurement and hyperkalemia in hospitalised patients using serum potassium concentration increasing drugs

Keywords: potassium, hyperkalemia, drug-drug interactions, monitoring, clinical risk management, clinical decision support

Purpose

Although, drug-drug interactions (DDIs) between potassium increasing drugs (PIDs) are known risk factors for developing hyperkalemia, not much is known about their risk and management strategies during hospitalisation. This study describes the frequency of serum potassium measurement and of hyperkalemia for hospitalised patients using one or more PIDs, and determinants thereof.

Methods

Adult patients hospitalised in the University Medical Centre Utrecht in 2006-2008 were included in this cross sectional study. Frequency of serum potassium measurement and hyperkalemia were compared between patients using only one PID at a time (monotherapy group) and patients using two or more PIDs concomitantly (interaction group).

Determinants studied were renal failure, diabetes mellitus, use of diuretics, type of DDI, start of the PIDs within the hospital versus continued home medication, and medical speciality.

Results

Serum potassium was measured more frequently in the interaction group compared to the monotherapy group (67% vs. 56%; RR=1.19 (1.14-1.24)) and the risk of hyperkalemia was also increased in the interaction group (9.9% vs. 5.9% RR=1.7 (1.3-2.1)). The combination of potassium sparing diuretics plus a potassium supplement, start of the PID within the hospital and hospitalisation at non-internal departments was associated with higher relative risk estimates for hyperkalemia.

Conclusions

Even when physicians receive a direct pop-up to monitor serum potassium levels when prescribing two PIDs concomitantly, serum potassium levels are not measured in 33% of patients and 10% of patients develop hyperkalemia.

Improved management strategies and/or clinical decision support systems are needed to decrease the frequency of hyperkalemia following DDIs.

Introduction

Drug-drug interactions (DDI) often lead to preventable adverse drug events and contribute to the burden of drug-related health damage [1,[START_REF] Juurlink | Drug-drug interactions among elderly patients hospitalized for drug toxicity[END_REF]. The interaction between serum potassium concentration increasing drugs (PIDs) is one of the most frequently occurring DDI. It is present in 8-10% of all hospitalized and non-hospitalized patients [START_REF] Buurma | Clinical risk management in Dutch community pharmacies: the case of drug-drug interactions[END_REF][START_REF] Zwart-Van Rijkom | Frequency and nature of drug-drug interactions in a Dutch university hospital[END_REF].

Hyperkalemia is a serious and potentially life threatening electrolyte disorder that follows from a disbalance in potassium homeostasis [START_REF] Perazella | Hyperkalemia in the elderly: drugs exacerbate impaired potassium homeostasis[END_REF]. The magnitude of the risk of hyperkalemia caused by combining two or more PIDs was highlighted after the findings of the RALES study were implemented in regular patient care. In 1999, the Randomized Aldactone Evaluation Study (RALES) demonstrated significantly improved outcomes for congestive heart failure patients treated with spironolactone in addition to treatment with diuretics and ACE-inhibitors [START_REF] Pitt | Effectiveness of spironolactone added to an angiotensin-converting enzyme inhibitor and a loop diuretic for severe chronic congestive heart failure (the Randomized Aldactone Evaluation Study [RALES])[END_REF]. Five years later, in 2004, it was found that after this publication prescription rates for spironolactone had clearly increased and that there was also a marked increase in hyperkalemia-associated morbidity and mortality [START_REF] Juurlink | Rates of hyperkalemia after publication of the Randomized Aldactone Evaluation Study[END_REF]. The risk does not only exist for renin-angiotensin system (RAS) inhibitors and spironolactone, but also for other combinations of PIDs including potassium supplements [START_REF] Cruz | Hyperkalaemia in congestive heart failure patients using ACE inhibitors and spironolactone[END_REF][START_REF] Desai | Hyperkalemia associated with inhibitors of the renin-angiotensin-aldosterone system: balancing risk and benefit[END_REF][START_REF] Henz | Influence of drugs and comorbidity on serum potassium in 15 000 consecutive hospital admissions[END_REF][START_REF] Johnson | Predicting the risk of hyperkalemia in patients with chronic kidney disease starting lisinopril[END_REF].

Many studies have investigated the risk to develop hyperkalemia from DDIs between PID"s, but most studies have been performed in an outpatient setting, upon hospital admission, in a specific patient group or with a specific combination of PID"s. Moreover, in these studies, serum potassium levels were often measured per protocol and were closely monitored. No published data are available about the risk and management strategies of combining PIDs during hospitalisation where the average stay is only a few days and the access to laboratory measurement is more easy.

Therefore, the objective of this study is to describe the frequency of serum potassium measurement and of hyperkalemia for hospitalised patients using one or more PIDs, and determinants there of.

Methods

Setting

This cross-sectional study was performed in the University Medical Centre Utrecht (UMCU), a 1042-bed academic medical centre located in the centre of the Netherlands. In 2006-2008, approximately 30,000 clinical hospitalisations took place annually. All medications for hospitalised patients are prescribed using a computerized physician order entry (CPOE) system. In the Netherlands, a working group of the Scientific Institute of Dutch Pharmacists developed and maintains an evidence based and professional guideline for the management of DDIs, the G-standard, described in detail elsewhere [START_REF] Van Roon | Clinical relevance of drug-drug interactions : a structured assessment procedure[END_REF]. For clinical risk management purposes, the G-standard is incorporated in the CPOE system [START_REF]Z-index[END_REF].

All drugs in the G-standard are coded according to the Anatomical Therapeutical Chemical Classification (ATC) index. The ATC code system is an international standard for drug utilization studies determined by a WHO International Working Group [START_REF]Anatomical Therapeutic Chemical (ATC) Classification Index[END_REF]. Every DDI in this guideline was assessed by four core parameters: (i) evidence on the interaction;

(ii) clinical relevance of the potential adverse reaction resulting from the interaction; (iii) risk factors identifying patient, medication or disease characteristics for which the interaction is of special importance; and (iv) the incidence of the adverse reaction. Based on the information in the G-standard, the CPOE generates an alert when a combination of two or more interacting drugs are prescribed concomitantly. In case of prescribing two PIDs concomitantly, the physician receives a direct pop-up that there is a risk of hyperkalemia and that the serum potassium levels should be monitored closely.

For research purposes all prescriptions are routinely exported in the Utrecht Patient Oriented Database (UPOD). UPOD is an infrastructure of relational databases comprising data on patient demographics, hospital discharge diagnoses, medical procedures, medication orders and laboratory tests for all patients treated at the UMCU since 2004, and has been described in detail elsewhere [START_REF] Berg | Linking laboratory and medication data: new opportunities for pharmacoepidemiological research[END_REF].

Study population

All patients aged 18 years or older hospitalised in the UMCU in 2006-2008 for ≥24 h and who were prescribed at least one PID were included. If a patient was admitted during the study period to the hospital more than once, only the first admission was included. Patients admitted to the dialysis department were excluded because patients at end-stage renal disease lack the renal compensation mechanism for potassium homeostasis and large variations in serum potassium levels may occur due to dialysis. Patients admitted to the intensive care (IC) units were excluded as well, because in the IC units a different CPOE system is used and the prescription data are not (yet) stored in the UPOD database.

Exposure

Patients who were prescribed one PID (monotherapy group) were compared to patients who were prescribed two or more PIDs concomitantly (interaction group). In this study, PIDs were defined as: a) renin-angiotensin system (RAS) inhibitors, b) potassium sparing diuretics or c) potassium supplements. The professional DDI clinical risk management guideline has defined a combination of these drugs as a risk for developing hyperkalemia. Other combinations of potentially PIDs mentioned in literature such as nonsteroidal anti-inflammatory drugs, β-adrenergic blocking drugs, heparin and trimethoprim are not considered relevant enough by the professional DDI clinical risk management guideline and were therefore not included.

Outcome

Two different outcome measures were studied: 1) frequency of serum potassium measurement and 2) frequency of hyperkalemia. Frequency of serum potassium measurement was defined as the percentage of patients for which a serum potassium level was measured at least once during the hospitalisation period. Frequency of hyperkalemia was defined as the percentage of patients with any serum potassium measurement ≥ 5.5 mEq/L during the hospitalisation period.

Determinants

In addition to age and gender, the following determinants were studied:

-Renal function: glomular filtration rates (eGFR) were estimated with the Modification of diet in renal disease (MDRD) equation [START_REF] Levey | A more accurate method to estimate glomerular filtration rate from serum creatinine: a new prediction equation. Modification of Diet in Renal Disease Study Group[END_REF][START_REF] Levey | Using standardized serum creatinine values in the modification of diet in renal disease study equation for estimating glomerular filtration rate[END_REF] using the highest serum creatinine level measured during the hospitalisation period.

A decreased renal function is a known risk factor for hyperkalemia.

-Diabetes mellitus: a patient was considered to have diabetes mellitus when one or more medication orders for any anti-diabetic medication had been prescribed during hospitalisation. Anti-diabetic medication was defined as medication with an ATC code "A10". Diabetes mellitus is also a known risk factor for hyperkalemia.

-Diuretics: any use of diuretics. This was defined as the use of lis-or thiazide diuretica or a combination of drugs containing thiazide diuretics during hospitalisation (ATC C03C, C03A). This determinant was studied, as both the use of thiazide-and lisdiuretics have been associated with a protective effect against the development of hyperkalemia [START_REF] Weinberg | Risk of hyperkalemia in nondiabetic patients with chronic kidney disease receiving antihypertensive therapy[END_REF] -Type of combination: in the interaction group three combinations are possible, namely 1) RAS-inhibitor + potassium sparing diuretic, 2) RAS-inhibitor + potassium supplement, 3) potassium sparing diuretic + potassium supplement.

-Start at home or start during hospitalisation: In the monotherapy group start at home was defined as the first PID prescribed in the CPOE system within 24 hours of arrival in the hospital. In the interaction group start at home was defined as the second PID prescribed in the CPOE system within 24 hours of arrival in the hospital.

PIDs prescribed more than 24 hours after admission were considered to be newly started within the hospital.

-Department: Internal departments versus non-internal departments. Internal departments were defined as general internal departments, Heart & lung departments and department of Geriatrics, Nephrology and Oncology. Non-internal departments consisted mainly of Surgery, Neurology, Psychiatry and Gynaecology departments.

Data analysis

Relative risks with corresponding 95% confidence intervals (CI) were calculated for both outcome measures. SPSS Package for Social Sciences software (SPSS) version 15.1 for windows (SPSS Inc., Chicago, IL) was used to analyze the collected data.

Results

In total 9,441 patients using at least one PID were included in this study of which 1,396 patients (14.7%) used two or more PIDs concomitantly at any time during their hospitalisation (interaction group). On average, patients using ≥2 PIDs were older, had a worse kidney function, and used anti-diabetic drugs and diuretics more often than patients using only one PID (monotherapy group) (Table 1). For patients in the monotherapy group, the PID was started within the hospital in 35.5% of patients and for the patients using > 2 PIDs, the second PID was first prescribed within the hospital in 50.4% of patients.

Serum potassium was measured more frequently in the interaction group compared to the monotherapy group (67% vs. 56% RR=1. [START_REF] Einhorn | The frequency of hyperkalemia and its significance in chronic kidney disease[END_REF] (1.14-1.24)) and the risk of hyperkalemia was also increased in the interaction group (9.9% vs. 5.9% RR=1.7 (CI 1.3-2.1)) (Table 2). For patients whom potassium was measured at least once, a median of 0.67 measurements per hospital admission day were performed in the interaction group and 0.50 in the monotherapy group.

When stratifying for age, gender, renal function, diabetes and use of diuretics, no differences were found in relative risk estimates between the monotherapy and interaction groups. The absolute risk of hyperkalemia however, was higher in patients with a eGFR ≤ 50 mL/min (p<0.01 for both monotherapy and interaction groups).

When comparing the risk of hyperkalemia for the different interaction types, the risk was found to be higher for the combination of a potassium supplement plus a potassium sparing diuretic (RR=3.0 (2.0-4.4)) than for the two other combinations: a potassium sparing diuretic plus RAS-inhibitor (RR=1.5 (1.1-2.1)) and a potassium supplement plus RAS-inhibitor (RR=1.3 (0.9-2.0)).

The relative risk estimate was also found to be higher when the PIDs were first prescribed during hospitalisation (RR=2.14 (1.60-2.88)) compared to when PIDs were already started at home (RR=1.13 (0.77-1.66)). Serum potassium levels were measured more frequently when PIDs were started during hospitalisation in both the interaction (63.0% vs 70.2%, p<0.01) and the monotherapy group (52.1% vs 63.6%, p<0.01).

When stratifying for type of department, the relative risk estimate for developing hyperkalemia was found to be higher for patients hospitalised at non-internal departments (RR=2.7 (1.7-4.3)) compared to patients hospitalised at internal departments (RR=1.3 (1.0-1.6)). The frequency of serum potassium measurement was higher for the patients hospitalised at internal departments, in both monotherapy (66.4% vs 47.5%, p<0.01) and interaction groups (68.6% vs 62.1%, p=0.018).

Discussion

In our hospital, serum potassium levels were measured in only 56-67% of patients using one or more PIDs.

Hyperkalemia, defined as serum potassium concentration ≥ 5.5 mEq/L, occurred in 6-10% of patients. The absolute risk for developing hyperkalemia was highest for patients with a eGFR≤ 50 and for patients with diabetes mellitus. However the relative risk estimates for comparing interaction to monotherapy were not increased for age, gender, kidney function and diabetes. When using two or more PID"s concomitantly, the risk of hyperkalemia was elevated in patients using a combination of a potassium sparing diuretic with a potassium supplement, in patients starting their PID therapy during hospitalisation and in patients hospitalised at non-internal departments.

A higher frequency of serum potassium measurement was found for the interaction group as compared to the monotherapy group. This may suggest that the interaction alerts, namely direct pop-ups advising the physicians to monitor potassium levels, were indeed (partly) effective. However, the higher frequency can also be caused by the longer duration of hospitalisation in the interaction group and differences in patients characteristics between the interaction group and monotherapy group. To check the assumption whether the higher frequency of serum potassium measurement was caused by an interaction signal, we also compared the interaction and the monotherapy group for the measurement of two "neutral markers", that is leukocyte count and haemoglobin level. We found that both markers were measured more frequently in the interaction group than in the monotherapy group (66% versus 58%, p < 0.01 for haemoglobin level, and 63% versus 51%, p < 0.01 for leukocyte count, results not shown). This means that the higher frequency of potassium measurement in the interaction group may not be (entirely) due to warnings about the drug-drug interaction, but may also be explained by differences in patient characteristics between patients using only one PID and patients using ≥2 PIDs. Apparently, more laboratory measurements were required in the interaction group, which may suggest that these patients were more severely ill or did have more complex diseases or more co-morbidity.

From Table 2 it can also be read that serum creatinine and serum potassium are always almost measured simultaneously. When renal function is known, serum potassium is measured in 97-99% of patients, while if renal function is not known serum potassium is measured in only about 3% of patients. It is likely that the same holds true for the measurement of for example sodium. As such, serum potassium measurement might not always be carried out with the intention to monitor potassium, it can also be carried out as an adjunct only.

As was to be expected, the frequency of hyperkalemia was much higher in patients with an estimated eGFR ≤ 50 mL/min than in patients with an estimated eGFR > 50 mL/min. It is known that patients with chronic renal failure are at risk developing hyperkalemia [START_REF] Einhorn | The frequency of hyperkalemia and its significance in chronic kidney disease[END_REF]. It is also known that the use of a PID may attribute to this risk in patients with renal failure [START_REF] Bakris | ACE inhibition or angiotensin receptor blockade: impact on potassium in renal failure. VAL-K Study Group[END_REF]. This risk however, did not further increase for patients using ≥ 2 PIDs (15.8% for the monotherapy group and 16.9% for the interaction group). Possibly, monotherapy already requires the maximum capacity of renal autoregulation. If so, this may not leave much space for a further increase in risk by the use of ≥ 2 PIDs concomitantly.

A comparable result was seen in diabetic patients: diabetes is a known risk factor for hyperkaliemia [START_REF] Henz | Influence of drugs and comorbidity on serum potassium in 15 000 consecutive hospital admissions[END_REF][START_REF] Einhorn | The frequency of hyperkalemia and its significance in chronic kidney disease[END_REF][START_REF] Stevens | Hyperkalemia in hospitalized patients[END_REF]. In our study however, the risk did not increase by the introduction of a second PID. Apperently risk factors to develop hyperkalemia DDI are not the same as risk factors to develop hyperkalemia after a DDI.

The frequency of hyperkalemia in patients using a thiazide diuretic seems to be lower than in patients without diuretics, which is suggestive for a protective effect in both monotherapy and interaction groups. This protective effect was not seen for the use of loop diuretics or for the combination of a thiazide and a loop diuretic in the co-medication. The use of diuretics is a known protective factor for the development of hyperkalemia [START_REF] Weinberg | Risk of hyperkalemia in nondiabetic patients with chronic kidney disease receiving antihypertensive therapy[END_REF]. Weinberg found a reduction in the probability of hyperkalemia of 59% for the use of diuretics. However, thiazide diuretics, not loop diuretics, were found to contribute the most to this effect [START_REF] Henz | Influence of drugs and comorbidity on serum potassium in 15 000 consecutive hospital admissions[END_REF].

When comparing the different combinations of PID"s, the highest risk was found for hospitalised patients using potassium sparing diuretics combined with potassium supplements. Of course, the risk of hyperkalemia is obvious for this combination. Some have even argued that an alert is superfluous in this situation [START_REF] Ponce | Drug-induced hyperkalemia[END_REF]. Indermitte et al found that the use of potassium supplements contributed most strongly to the velocity of developing hyperkalemia [START_REF] Weinberg | Risk of hyperkalemia in nondiabetic patients with chronic kidney disease receiving antihypertensive therapy[END_REF]. Others, however, found that a combination with RAS-inhibitors was the strongest predictor to develop hyperkalemia in hospitalised patients [START_REF] Henz | Influence of drugs and comorbidity on serum potassium in 15 000 consecutive hospital admissions[END_REF].

Patients starting interacting PIDs during hospitalisation were at higher risk developing hyperkalemia compared to patients who started their interacting PID therapy at home. This may be explained by the assumption that, when the interacting medication was started already at home, serum potassium levels may have been monitored before the patient was hospitalised. Adjustments may have been made, for example by decreasing the dose of the PID"s or adding potassium serum level decreasing medication [START_REF] Vereijken | Risk calculation for hyperkalaemia in heart failure patients[END_REF]. Another assumption may be that PIDs cause only a temporary rise of serum potassium levels. Moreover, one might expect that physicians are more inclined to monitor the effects of newly added medication than of continued home-medication [START_REF] Van Der Sijs | Drug safety alert generation and overriding in a large Dutch university medical centre[END_REF]. Indeed, the serum potassium levels were measured somewhat more often in the group of patients who started the PIDs in the hospital, than in the group of patients who continued their PID home-medication. This was seen in both monotherapy and interaction group (63.6% vs 52.1%, p< 0.01 and 70.2% vs 63.0%, p< 0.01 respectively).

Finally, the relative risk estimate for developing hyperkalemia was higher for patients hospitalised at non-internal departments (RR=2.7 (1.7-4.3)) compared to patients hospitalised at internal departments (RR=1.3 (1.0-1.6)). The frequency of serum potassium measurement was also increased for patients with an interaction (47.5% vs 62.1% RR=1.3 (1.2-1.4)) but the absolute frequency of serum potassium measurement was remarkably low for patients with monotherapy at the non-internal departments. This may again suggest that warning signals were followed. The frequency of the measurement of the two neutral markers however, were also different (54.2% vs 40.2%, p< 0.01 for leucocytes and 62.3% vs 50.3%, p< 0.01 for haemoglobin) which means that differences in patient characteristics may have played a role as well. Moreover the frequency of serum potassium measurement in the monotherapy group of the non-internal departments is relatively low. This may reflect that physicians working at non-internal departments are less preoccupied with serum electrolyte levels as physicians working at internal departments. As such, they may be less aware of the risk for the development of hyperkalemia when prescribing a PID. Therefore, special attention is needed for patients using PIDs at non-internal departments.

One of the potential limitations of this study may be that nonsteroidal anti-inflammatory drugs, β-adrenergic blocking drugs, heparin and trimethoprim were not included in our definition of a PID. This was done on purpose, as they increase serum potassium levels to a far lesser extent than RAS-inhibitors, potassium sparing diuretics and potassium supplements. As such, these drug combinations are not included in the G-standard as DDIs and the CPOE does not generate a pop-up to warn for hyperkalemia.

Second, the local hospital setting may limit the generalisability of the results.

Finally, as this is an observational study, patients were not randomised to either the monotherapy or the interaction group and serum potassium levels were not measured per protocol. The differences in baseline characteristics make it very clear that the patients in the monotherapy and the interaction group are not the same. In addition, serum potassium levels were not measured systematically in all patients, but were known for only 56-66% of patients, meaning that there is a testing bias [START_REF] Velthove | Testing bias in clinical databases: methodological considerations[END_REF]. As such, this study does not aim to answer the etiologic question about the contribution of DDIs to the risk of developing hyperkalemia. The objective was to study the consequences of a frequently encountered DDI in a daily-hospital practice and to gain insight in the determinants thereof. As such, adjusted relative risk estimates have consciously been omitted from Table 2, and only uni-variate analysis have been performed. The message is not so much that these DDIs cause hyperkalemia; but that, in daily practice, patients in the interaction group deserve extra attention and serum potassium measurement.

In our hospital, when prescribing two PIDs concomitantly, the physicians receive a direct pop-up that there is a risk of hyperkalemia and that the serum potassium levels should be monitored closely, however this advice was not followed in 33% of patients. One of the reasons for this may be that the warnings are not always appropriate. The DDI signal always appears when a DDI occurs, even if the serum potassium level is actually low. This may cause alert fatigue [START_REF] Van Der Sijs | Overriding of drug safety alerts in computerized physician order entry[END_REF][START_REF] Weingart | Physicians' decisions to override computerized drug alerts in primary care[END_REF].

Another reason may be that the alert to monitor potassium serum levels only occurs when the medication is started.

There are no reminders later in time, while in fact it will take at least a few days before a rise in serum potassium levels will be revealed.

More advanced methods of clinical decision support may be suitable to improve the management of this DDI and to reduce the frequency of hyperkalemia. For example, it may be worthwhile to generate a reminder every 3-4 days to monitor serum potassium for all hospitalised patients using two or more PIDs. Another possibility would be to link laboratory and medication data and to generate a warning when serum potassium levels are increasing above a certain limit for patients using one or more PIDs.

Table 1 Baseline characteristics for monotherapy group (users of 1 potassium increasing drug) versus interaction group (users of ≥2 PIDs)

 1

		Monotherapy:	Interaction:	p value a
		1 potassium increasing	≥2 potassium increasing
		drug	drugs
		N=8,045	N=1,396
	Length of stay (median days	6.18 (3.0-12.0)	8.9 (4.2-16.1)	<0.001
	(interquartile range)
	Age (mean (95% CI))	62.2 (61.8-62.5)	67.3 (66.6-68.1)	<0.001
	Gender (Male %)	3,981 (49.5%)	751 (53.8%)	<0.01
	Renal function			<0.001
	-eGFR ≥80 mL/min	1,479 (18.4%)	123 (8.8%)
	-eGFR 50-80 mL/min	1,876 (23.3%)	375 (26.9%)
	-eGFR ≤50 mL/min	1,150 (14.3%)	422 (30.2%)
	-Unknown	3,540 (44.0%)	476 (34.1%)
	Diabetes mellitus	1,582 (19.7%)	340 (24.4%)	<0.001
	Diuretics	3,106 (38.6%)	1,097 (78.6%)	<0.001
	Drug type	First MO		n.a.
	-RAS inhibitor	4,819 (59.9%)
	-potassium sparing diuretic	622 (7.7%)
	-potassium supplement	2,604 (32.4%)
	Interaction type			n.a.
	-RAS-inhibitor + potassium sparing		707 (50.6%)
	diuretic		501 (35.9%)
	-RAS-inhibitor + potassium		188 (13.5%)
	supplement
	-potassium sparing diuretic +
	potassium supplement
	Start drug/interaction			<0.001
	-at home	5,192 (64.5%)	692 (49.6%)
	-during hospitalisation	2,853 (35.5%)	704 (50.4%)
	Department			<0.001
	-internal medicine specialities	3,694 (45.9%)	977 (70.0%)
	-non-internal medical specialities	4,351 (54.1%)	419 (30.0%)
	eGFR, estimated glomerular filtration rate; RAS, renin-angiotensin system, n.a., not applicable

a) Chi square test (categoral variables) or t-test (continuous variables) as appropriate.

Table 2 Percentage of patients in whom serum potassium has been measured and percentage of patients with hyperkalemia (defined as serum potassium level ≥ 5.5 mEql/L) for users of 1 potassium increasing drug (PID) versus users of ≥2 PIDs Potassium measured Hyperkalemia

 2

		Monotherapy	Interaction	RR (95% CI)	Monotherapy	Interaction	RR (95% CI)
		1 potassium increasing	≥2 potassium		1 potassium	≥2 potassium
		drug	increasing drugs		increasing drug	increasing drugs
		(n=8,045)	(n=1,396)		(n=4,520)	(n=930)
	Overall	56.2% (4,520/8,045)	66.6% (930/1,396)	1.19 (1.14-1.24)	5.9% (267/4,520)	9.9% (92/930)	1.68 (1.34-2.10)
	Age
	18-50 y	56.8% (989/1,742)	70.8% (114/161)	1.25 (1.12-1.39)	6.2% (61/989)	12.3% (14/114)	1.99 (1.15-3.44)
	50-70 y	55.3% (1,921/3,473)	64.3% (369/574)	1.16 (1.09-1.24)	5.6% (107/1,921)	8.4% (31/369)	1.51 (1.03-2.21)
	70-80 y	54.7% (964/1,761)	68.5% (263/384)	1.25 (1.16-1.36)	5.2% (50/964)	9.5% (25/263)	1.83 (1.16-2.90)
	>= 80 y	60.4% (646/1,069)	66.4% (184/277)	1.10 (1.0-1.21)	7.6% (49/646)	12.0% (22/184)	1.58 (0.98-2.54)
	Gender
	male	58.4% (2,326/3,981))	67.4% (506/751)	1.15 (1.09-1.22)	6.3% (147/2,326)	9.5% (48/506)	1.50 (1.10-2.05)
	female	54.0% (2,194/4,064)	65.7% (424/645)	1.22 (1.14-1.30)	5.5% (120/2,194)	10.4% (44/424)	1.90 (1.37-2.64)
	Renal function
	-eGFR ≥80 mL/min
	-eGFR 50-80 mL/min
	-eGFR ≤50 mL/min
	-Unknown

Competing interests

The authors do not have a potential conflict of interest. The Department of Pharmacoepidemiology and Pharmacotherapy employing authors W.W.v.S. and A.C.G.E. have received unrestricted funding for pharmacoepidemiological research from GlaxoSmithKline, Novo Nordisk, the private-public-funded Top Institute Pharma (http://www.tipharma.com,includes co-funding from universities, government, and industry), the Dutch Medicines Evaluation Board, and the Dutch Ministry of Health.

The authors are grateful to Hanneke den Breeijen for the data analysis and to their colleagues at the Utrecht Institute for Pharmaceutical Sciences and the UMC Utrecht for their support in establishing and maintaining UPOD.


V Bonnaillie-Noël

M Dambrine

Interactions between moderately close circular inclusions: the Dirichlet-Laplace equation in the plane

Introduction

The presence of small inclusions or of a surface defect modifies the solution of the Laplace equation posed in a reference domain Ω 0 . If the characteristic size of the perturbation is small, then one can expect that the solution of the problem posed on the perturbed geometry is close to the solution of the reference shape. Asymptotic expansion with respect to that small parameter -the characteristic size of the perturbation-can then be performed.

The case of a single inclusion ω, centered at the origin 0 being either in Ω 0 or in ∂Ω 0 , has been deeply studied, see [START_REF] Maz ′ Ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF][START_REF] Maz ′ Ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF][START_REF] Il ′ In | Matching of asymptotic expansions of solutions of boundary value problems[END_REF][START_REF] Lewiński | Topological derivative for nucleation of non-circular voids. The Neumann problem[END_REF][START_REF] Nazarov | Asymptotic analysis of shape functionals[END_REF][START_REF] Dambrine | Influence of a boundary perforation on the Dirichlet energy[END_REF][START_REF] Dambrine | A multiscale correction method for local singular perturbations of the boundary[END_REF]. The techniques rely on the notion of profile, a normalized solution of the Laplace equation in the exterior domain obtained by blow-up of the perturbation. It is used in a fast variable to describe the local behavior of the solution in the perturbed domain. In usually dealt situations that is for Neumann boundary conditions in dimension greater than two and Dirichlet boundary condition in dimension greater than three, convergence of the asymptotic expansion is obtained thanks to the decay of the profile at infinity.

The case of several inclusions was considered for example in the series of papers of A. Movchan and V. Maz'ya [START_REF] Maz'ya | Uniform asymptotic formulae for Green's functions in singularly perturbed domains[END_REF][START_REF] Maz'ya | Asymptotic treatment of perforated domains without homogenization[END_REF] where an asymptotic approximation of Green's function is built and justified in a domain with many inclusions. The points where the inclusion is dug are fixed in those works. In [START_REF] Bonnaillie-Noël | Interactions between moderately close inclusions for the Laplace equation[END_REF], the Neumann case where the distance between the holes tends to zero but remains large with respect to their characteristic size was investigated for two perfectly insulated inclusions: a complete multiscale asymptotic expansion of the solution to the Laplace equation is obtained in a three scales case.

We consider in the present work the case of two defects with Dirichlet boundary conditions in a bidimensional domain. For the simplicity of the presentation, we assume that the defects we are considering are disks. Let Ω 0 be a bounded domain of R 2 containing the origin 0. For ε > 0, small enough, we define the perturbed domain Ω ε as

Ω ε = Ω 0 (ω - ε ∪ ω + ε ), with ω ± ε = x ± ε + εB(0, 1), (1.1)
where x ± ε = ±η ε d with a given unitary vector d, and a real number η ε . Shortly, Ω ε consists of Ω 0 from which two ε-disks at distance 2η ε have been removed. We aim at building an asymptotic expansion of the solution u ε of the Laplace problem in Ω ε :

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -∆u ε = f in Ω ε , u ε = 0 on Γ = ∂Ω 0 , ∂ n u ε = 0 on ∂ω ± ε , (1.2)
for some L 2 datum f whose support does not contain the origin 0. We will consider two unstudied cases: η ε is constant and η ε = ε α for α ∈ (0, 1):

• In the first case, we are considering two small holes of size ε around two fixed points A and B and the distance between both is hence fixed. For the cases of Neumann boundary condition or of Dirichlet boundary conditions in dimension at least three, this cases can be treated by separating each hole through cut-off functions and hence reducing it to the single inclusion case. Here, the presence of the logarithmic term prohibits this approach and the interaction between the holes has to be studied.

• In the second case, the distance between A and B collapses to 0 with ε like ε α , that is to say slower than the size ε of the inclusions. The interaction between the two holes are then stronger and we will prove that the leading order of the asymptotic expansion is then modified.

This work is hence organized as follows. In a first section, we consider the well studied case of a single inclusion. We explain why the usual method fails in this case, we reformulate ideas of presented in [START_REF] Maz ′ Ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF][START_REF] Maz ′ Ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF][START_REF] Maz'ya | Uniform asymptotic formulae for Green's functions in singularly perturbed domains[END_REF] in order to explain both the strategy we shall follow in the cases of two inclusions and the difficulties we will have to face. In a second step, we consider the case of two defects around two fixed points. We then consider the case of weakly interacting defects: that is to say the distance between the centers of the defects is not fixed but varies like ε α with the small parameter ε, we show that the structure of the expansion is modified by this interaction.

Single inclusion case 2.1 Default of the usual method

We consider a smooth bounded domain Ω 0 such that the origin is contained in Ω. Let f ∈ L 2 (Ω 0 ) be such that 0 ∉ suppf . For ε small enough such that εB(0, 1) ∩ suppf = ∅, we define a perturbed domain

Ω ε = Ω 0 ω ε with ω ε ∶= εB(0, 1).
We are interested in describing u ε the solution of

-∆u ε = f in Ω ε , u ε = 0 on ∂Ω ε . (2.1)
Since the H 1 -capacity of ω ε tends to 0 as ε tends to 0 (like 1 ln ε), the function u ε tends to u 0 defined as the solution in the unperturbed domain:

-∆u 0 = f in Ω 0 , u 0 = 0 on ∂Ω 0 . (2.2)
To catch the asymptotic of u ε , let us study the error r 0

ε ∶= u ε -u 0 . It solves ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -∆r 0 ε = 0 in Ω ε , r 0 ε = 0 on ∂Ω 0 , r 0 ε = -u 0 on ∂ω ε .
The solution u 0 of the unperturbed problem (2.2) is analytic inside Ω 0 . Hence, it can be approximated in the vicinity of the origin by its Taylor expansion:

u 0 (x) = u 0 (εX) = u 0 (0) + k≥1 D k u 0 (0)[X, . . . , X] ε k k! , ∀x = εX ∈ ∂ω ε .
Each term of this expansion can be seen as an error of order k: the usual idea to improve the expansion of u ε is now to lift in an harmonic way each term. In the 3-dimension case or for Neumann boundary conditions, this asymptotics analysis is well studied (see for example [START_REF] Maz ′ Ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF][START_REF] Maz ′ Ya | Asymptotic theory of elliptic boundary value problems in singularly perturbed domains[END_REF][START_REF] Bonnaillie-Noël | Interactions between moderately close inclusions for the Laplace equation[END_REF]...). The key point is the notion of profile, a normalized solution of the Laplace equation in the exterior domain obtained by blow-up of the perturbation. It turns out that profiles tend to 0 at infinity at least in the above mentioned cases. In the converse, for the case considered here (Dirichlet condition in dimension 2), the crucial property is not satisfied: it supposes that the boundary value problem

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -∆V = 0 in R 2 ω, V = u 0 (0) on ∂ω, V → 0 at infinity, (2.3)
has a solution. Here ω is a smooth bounded domain of R 2 . Unfortunately, this is not the case except when u 0 (0) = 0. The classical analysis of elliptic equation in unbounded domain is made in the functional setting of weighted Sobolev spaces, see [START_REF] Amrouche | Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator: an approach in weighted Sobolev spaces[END_REF]. It is known that (2.3) has a unique solution in a space containing the constants, hence this solution is the constant u 0 (0) which prohibits the condition at infinity if u 0 (0) ≠ 0.

The existence, uniqueness and behavior at infinity of the solution for a general boundary condition in H 1 2 (∂ω) is given by the following Lemma.

Lemma 2.1 1. Let ω be a smooth bounded domain of R 2 with 0 ∈ ω. Let f ∈ H 1 2 (∂ω). Then the boundary value problem -∆V = 0 in R 2 ω, V = f on ∂ω, (2.4)
admits a unique weak solution V in the variational space

W 1,2 0 (R 2 ω) = u; u (1 + X ) ln(2 + X ) ∈ L 2 (R 2 ω) and ∇u ∈ L 2 (R 2 ω) .
2. Furthermore, the solution V can be decomposed

V (X) = V 0 + n k=1 V k (X) + O ∞ ( X -n+1 ),
where V 0 is a constant and V k (X) = V k (θ) X -k where θ is the usual polar angle and V k ∈ Span(cos kθ, sin kθ).

3.

If ω is the unit ball B(0, 1), and if ∫ ∂B(0,1) f = 0, then the solution V reads

V (X) = n k=1 V k (X) + O ∞ ( X -n+1 ) with V k (r, θ) = a k (f ) cos kθ + b k (f ) sin kθ r k .
In particular, there exists a constant C such that

V (X) ≤ C X and ∇V (X) ≤ C X 2 .
Proof:

1. This result is proved by Giroire in [START_REF] Giroire | Étude de quelques problèmes aux limites extérieurs et résolution par équations intégrales[END_REF].

2. The solution V has a trace on the circle ∂B(0, R) such that ω ⊂ B(0, R). Let us decompose this trace in Fourier series:

v(θ) = a 0 (v) + k≥1 a k (v) cos kθ + b k (v) sin kθ .
The restriction of V to the complement R 2 B(0, R) solves

-∆V = 0 in R 2 B(0, R), V = v on ∂B(0, R),
and hence reads

V (r, θ) = a 0 (v) + k≥1 a k (v) cos kθ + b k (v) sin kθ R k r k .
This satisfies the announced statement.

3. It suffices to check that a 0 (v) = 0.

First term in slow variable with the logarithmic solution

We are in the situation where the boundary value problem (2.3) defining the corrector has in general no solution. To circumvent this obstruction, the logarithmic solution has to be considered. However, it tends to infinity at infinity, it is not of finite energy in R 2 ω and has to be considered only in Ω 0 . Its trace on ∂Ω 0 is of size one and has to be corrected.

That is why we introduce the function w defined as the solution of -∆w = 0 in Ω 0 , w = ln x on ∂Ω 0 .

(2.5)

The idea is now to combine the logarithmic solution and the lifted w to build a corrector incorporating the scales 1 and ε. To build this corrector, we search coefficients a(ε) and b(ε) such that r 1 ε defined by

u ε (x) = u 0 (x) + a(ε) ln x + b(ε)w(x) + r 1 ε (x), is reduced with respect to r 0 ε ∶= u ε -u 0 . The remainder r 1 ε satisfies ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -∆r 1 ε = 0 in Ω ε , r 1 ε (x) = -(a(ε) + b(ε)) ln x on ∂Ω 0 , r 1 ε (x) = -u 0 (x) -a(ε) ln x -b(ε)w(x) on ∂ω ε .
For x ∈ ∂Ω 0 , we have

r 1 ε (x) = O(1) ⇔ a(ε) + b(ε) = O(1).
For x ∈ ∂ω ε , there exists X ∈ B(0, 1) such that x = εX and we have

r 1 ε (x) = O(1) ⇔ u 0 (0) + a(ε) ln ε + w(0) b(ε) = O(1)
Hence we solve the linear system in the unknowns (a(ε), b(ε)):

a(ε) + b(ε) = 0, a(ε) ln ε + b(ε) w(0) = -u 0 (0), to set a(ε) = 1 w(0) -ln ε u 0 (0) and b(ε) = -1 w(0) -ln ε u 0 (0).
A new scale appears:

h ε = 1 w(0) -ln ε = -1 ln ε + O 1 ln 2 ε as ε → 0. (2.6)
We define the normalized corrector w as

w(x) = ln x -w(x). (2.7)
Let us remark that ∆w = 2πδ 0 in D ′ (Ω 0 ) and w = 0 on ∂Ω 0 . It is a type of Green function for the domain Ω 0 . Let us emphasize that this function w is defined on Ω ε and can not be defined in either Ω 0 , either R 2 ω, the limit domains where the usual correctors are defined: the usual correctors in the slow variable x are defined on Ω 0 while the correctors in the rescaled variable x ε are defined in R 2 ω. We approximate u ε by

u ε (x) = u 0 (x) + u 0 (0)w(x) h ε + r1 ε (x).

Second term in fast variable

We check that the remainder r1

ε satisfies ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -∆r 1 ε = 0 in Ω ε , r1 ε = 0 on ∂Ω 0 , r1 ε = -u 0 -u 0 (0)w h ε on ∂ω ε .
The functions u 0 and w are analytic in the vicinity of 0. Hence, for x = εX ∈ ∂ω ε , we have

r1 ε (x) = -u 0 (0) - k≥1 D k u 0 (0)[X, . . . , X] ε k k! -u 0 (0) ln ε + ln X -w(0) - k≥1 D k w(0)[X, . . . , X] ε k k! h ε = - k≥1 D k u 0 (0)[X, . . . , X] ε k k! -u 0 (0) ln X h ε + u 0 (0) k≥1 D k w(0)[X, . . . , X]h ε ε k k! .
There are two kinds of terms:

• the terms coming from Taylor expansions at the origin 0 of functions (u 0 , w) defined as solution of a Laplace boundary value problem posed in Ω 0 ,

• the term in ln X which provides information about the geometry of the inclusion ω.

All these terms are zero mean value on ∂ω ε . They can be lifted by using Lemma 2.1.

Since we made the assumption that ω is a unit ball B(0, 1), the term ln X cancels. Thus the remainder r1 ε reads for x = εX ∈ ∂ω ε :

r1 ε (x) = k≥1 u k (X) ε k + k≥1 w k (X) h ε ε k , with u k = - 1 k! D k u 0 (0)[X, . . . , X] and w k = u 0 (0) k! D k w(0)[X, . . . , X].
Each appearing term in the remainder r1 ε is a homogeneous harmonic polynomial. Hence the mean value property insures that ∫ ∂B(0,1) u k = 0 and ∫ ∂B(0,1) w k = 0. Then Lemma 2.1 (point 3) can be applied. We set U k,1 and W k,1 the solution of problem (2.4) with v = u k and v = w k respectively. These functions can be decomposed as

U k,1 (X) = k j=1 U j k,1 (X) with U j k,1 (r, θ) = a j (u k ) cos jθ + b j (u k ) sin jθ r j , (2.8)
W k,1 (X) = k j=1 W j k,1 (X) with W j k,1 (r, θ) = a j (w k ) cos jθ + b j (w k ) sin jθ r j . (2.9)
We then approximate u ε by

u ε (x) = u 0 (x) + u 0 (0)w(x) h ε + k≥1 U k,1 ( x ε ) ε k + k≥1 W k,1 ( x ε ) h ε ε k + r2 ε (x).

Third term in slow variable

We have now to study the remainder r2

ε . It satisfies ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ -∆r 2 ε = 0 in Ω ε , r2 ε = - k≥1 U k,1 ( x ε ) ε k - k≥1 W k,1 ( x ε ) h ε ε k on ∂Ω 0 , r2 ε = 0 on ∂ω ε .
(2.10)

According to decomposition (2.8) and (2.9), the traces T U j k,1 and

T W j k,1 on ∂Ω 0 of the functions U j k,1 ( x ε ) and V j k,1 ( x ε ) satisfy T U j k,1 H 1 2 (∂Ω 0 ) = O(ε j ) and T W j k,1 H 1 2 (∂Ω 0 ) = O(ε j ). Then, we get T U k,1 H 1 2 (∂Ω 0 ) = O(ε) and T W k,1 H 1 2 (∂Ω 0 ) = O(ε). (2.11)
We now lift these functions in slow variables and define u k,1 and w k,1 as the solution of the normalized problems:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -∆u k,1 = 0 in Ω 0 , u k,1 = -1 ε T U k,1 on ∂Ω 0 , and ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -∆w k,1 = 0 in Ω 0 , w k,1 = -1 ε T W k,1 on ∂Ω 0 .
We can read u ε :

u ε (x) = u 0 (x) + h ε u 0 (0)w(x) + k≥1 U k,1 ( x ε ) ε k + k≥1 W k,1 ( x ε ) h ε ε k + k≥2 u k-1,1 (x) ε k + k≥2 w k-1,1 (x) h ε ε k + r3 ε (x).

Antsatz of asymptotic expansion

We inject the remainder r3 ε in the equation and obtain:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ -∆r 3 ε = 0 in Ω ε , r3 ε = 0 on ∂Ω 0 , r3 ε = - k≥2 u k-1,1 ε k - k≥2 w k-1,1 h ε ε k on ∂ω ε .
The function u k,1 and w k,1 have a value at 0 which is a priori not 0. This suggests to mimic the approach already used to start the asymptotics and to define r4 ε by:

u ε (x) = u 0 (x) + u 0 (0)w(x) h ε + k≥1 U k,1 ( x ε ) ε k + k≥1 W k,1 ( x ε ) h ε ε k + k≥2 u k-1,1 (x) ε k + k≥2 w k-1,1 (x) h ε ε k + k≥2 u k-1,1 (0)w(x) h ε ε k + k≥2 w k-1,1 (0)w(x) h 2 ε ε k + r4 ε (x).
This justifies the antsatz

u ε (x) = u 0 (x) + u 0 (0)w(x) h ε + j≥1,k≥2 α j,k w(x) h j ε ε k + j≥0,k≥1 V j,k ( x ε ) h j ε ε k + j≥0,k≥2 v j,k (x) h j ε ε k , (2.12)
where any coefficient α j,k and any functions V j,k and v j,k are obtained by reiterating the previous steps.

Justification of the order of convergence

Now we have to justify this formal expansion by using an a priori error estimate. Let us introduce the remainder of the asymptotic expansion truncated after the order h

J ε ε K r J,K ε (x) = u ε (x) -u 0 (x) -u 0 (0)w(x) h ε - (j,k)∈K J,K ,j≥1,k≥2 α j,k w(x) h j ε ε k + (j,k)∈K J,K ,k≥1 V j,k ( x ε ) h j ε ε k + (j,k)∈K J,K ,k≥2 v j,k (x) h j ε ε k , with K J,K = {(j, k) ∈ N 2 ; k < K or k = K and j ≤ J}.
By construction, r J,K ε is harmonic.

By the definition of the trace space of H 1 (Ω ε ) with the norm

f T H 1 (Ωε) = inf{ u H 1 (Ωε) ; u ∈ H 1 (Ω ε ) with u = f on ∂Ω ε }, we have r J,K ε H 1 (Ωε) = r J,K ε T H 1 (Ωε)
. The equality holds when u is harmonic since its achieves the infimum in the definition of the T H 1 norm. To overcome the difficulty to evaluate the T H 1 norm of a given function, we also define the intrinsic Sobolev space H 1 2 of the boundary of an ε-dependent domain Ω ε as a subspace of L 2 (∂Ω ε ) with finite norm

f H 1 2 (∂Ωε) = f L 2 (∂Ωε) + [f ] 2,∂Ωε , with [f ] 2 2,∂Ωε = ∬ ∂Ωε×∂Ωε f (x) -f (y) 2
xy 2 dσ x dσ y .

E. Gagliardo has shown in [START_REF] Gagliardo | Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabili[END_REF] that, if the domain is Lipschitz, the two different norms on H 1 2 are equivalent. For a familly of domains parametrized by ε, this means that the domains should be uniformly Lipschitz with respect to ε. In the case we are studying: a single interior perturbation Ω ε = Ω 0 ω ε where the nucleation center 0 belongs to Ω 0 , the uniform Lipschitz property is not satisfied and the norms H 1 2 and T H 1 may be non equivalent.

V. Maz'ya and S. Poborchi discuss in [START_REF] Maz ′ Ya | Differentiable functions on bad domains[END_REF]Section 4.1.3] situations where this property is violated. The two terms in the intrinsic norm should be weighted. We quote their result once adapted to the space H 1 2 in the case of plane domains: the trace norm f T H 1 (Ωε) is equivalent, uniformly in ε, to the norm

(ε ln ε ) -1 2 f L 2 (∂Ωε) + [f ] 2,∂Ωε . (2.13)
We have now to face the question of evaluating on ∂ωε various norm of the trace of a function f wich is smooth around 0. Quoting [4, Lemma 3.2], if f is a smooth function defined around 0, let M ≥ 0 be such that for all multi-indices k with

k < M , ∂ k f (0) = 0. Then, f H 1 2 (ε∂ω) ≤ Cε M , (2.14)
f T H 1 (εω) ≤ C ln ε -1 2 ε M -1 2 , (2.15)
with C independent of ε. Applying this to r J,K ε , we get

r J,K ε H 1 (Ωε) ≤ C h J-1 2 ε ε K-1 2 .
Remark 2.2 Let us conclude this section by noting that the usual a priori estimates can be applied for the second remainder r 2 ε directly since its trace on the non-uniformly Lipschitz boundary ∂ω ε is 0. Hence, we get directly from (2.10)-(2.11) that r 2 ε H 1 (Ωε) ≤ Cε 2 , that is:

u ε (x) = u 0 (x) + u 0 (0)w(x) h ε + U 1,1 ( x ε ) ε + W 1,1 ( x ε ) h ε ε + O H 1 (Ωε) (ε 2 ).

Numerical illustration

In this subsection, we illustrate the contribution of the first two terms of the expansion (2.12). For this, we choose for Ω 0 the rectangle [-1, 1] × [0, 1] and we consider the

f (x) = 2(x 2 2 -x 2 ) + 2(x 2 1 -1).
Then the solution u 0 of the problem (2.2) is given explicitely by

u 0 (x) = -(x 2 1 -1)(x 2 2 -x 2 ),
as illustrated in Figure 1. We compute the solution u ε of problem (2.1) by a finite element method using the Finite Element Library Mélina (see [START_REF] Martin | Mélina, bibliothèque de calculs éléments finis[END_REF]). The computed solution is given in Figure 2. The Figure 2: u ε difference r 0 ε = u εu 0 is given in Figure 3. We observe that that error is located around the inclusion since u 0 does not satisfy the Dirichlet condition on the inclusion.

Let us know observe the effect of the second term of the asymptotics. We compute an approximation of the function w, solution of (2.5) on Ω 0 by a finite element method, see Figure 4(a). The second term x ↦ u 0 (0)w(w) h ε = u 0 (0)(ln xw(x)) h ε of the expansion is computed in Figure 4(b). We add the first two terms of the expansion in Figure 5(a) with the scale h ε . We then compute the second remainder r1 ε = u εu 0u 0 (0)w h ε in Figure 5(b). We observe the error is now located between at the middle of Ω ε far away the boundary.

If we replace the asymptotics scale h ε by its first order expansion -1 ln ε, we change the second term of the expansion by the multiplicative term as illustrated in Figure 4(c). If we approximate h ε and -1 ln ε and then u ε by u 0u 0 (0)w ln ε as in Figures 6(a) and 6(b), we observe that the error if located around the inclusion. We highlight the interest of choosing the adapted scale h ε and not its first order approximation -1 ln ε. Table 1 gives the error norms when we aprroximate u ε by the first terms of its expansion. -

ε -u 0 u ε -u 0 -u 0 (0)w h ε u ε -u 0 + u 0 (0)w ln ε L 2 -
∆u ε = f in Ω ε , u ε = 0 on ∂Ω ε , (3.1)
set in the domain:

Ω ε = Ω 0 (ω A ε ∪ ω B ε ), with ω A ε = B(A, ε) and ω B ε = B(B, ε), (3.2)
as illustrated in Figure 7. Introduce the functions w A and w B solutions of

ω A ε ω B ε Ω ε Γ A • • B Figure 7:
The perturbed domain

-∆w A = 0 in Ω 0 , w A = ln x -A on ∂Ω 0 , and
-∆w B = 0 in Ω 0 , w B = ln x -B on ∂Ω 0 . (3.3)
We follow the previously introduced algorithm and seek coefficients a A (ε), b A (ε) and a B (ε), b B (ε) so that the remainder r 1 ε defined as

u ε (x) = u 0 (x) + a A (ε) ln x -A + b A (ε)w A (x) + a B (ε) ln x -B + b B (ε)w B (x) + r 1 ε (x), is smaller than r 0 ε = u ε -u 0 . The function r 1 ε satisfies ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ -∆r 1 ε = 0 in Ω ε , r 1 ε (x) = -(a A (ε) + b A (ε)) ln x -A -(a B (ε) + b B (ε)) ln x -B on ∂Ω 0 , r 1 ε (x) = -u 0 (x) -a A (ε) ln x -A -b A (ε)w A (x) -a B (ε) ln x -B -b B (ε)w B (x) on ∂(ω A ε ∪ ω B ε ).
The functions x ↦ ln x -A and x ↦ ln x -B defined on ∂Ω 0 are linearly independent since A ≠ B. Then for x ∈ ∂Ω 0 ,

r 1 ε (x) = O(1) ⇔ a A (ε) + b A (ε) = O(1) and a B (ε) + b B (ε) = O(1).
For x ∈ ∂ω A ε , there exists X ∈ B(0, 1) such that x = A + εX and we have

r 1 ε (x) = O(1) ⇔ u 0 (A) + a A (ε) ln ε + b A (ε) w A (A) + a B (ε) ln d + b B (ε) w B (A) = O(1).
By symmetry, for x ∈ ∂ω B ε , there exists X ∈ B(0, 1) such that x = B + εX and we have

r 1 ε (x) = O(1) ⇔ u 0 (B) + a A (ε) ln d + b A (ε) w A (B) + a B (ε) ln ε + b B (ε) w B (B) = O(1).
Hence we solve the linear system in the unknowns

(a A (ε), b A (ε), a B (ε), b B (ε)): ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a A (ε) + b A (ε) = 0, a B (ε) + b B (ε) = 0, a A (ε) ln ε + b A (ε) w A (A) + a B (ε) ln d + b B (ε) w B (A) = -u 0 (A), a A (ε) ln d + b A (ε) w A (B) + a B (ε) ln ε + b B (ε) w B (B) = -u 0 (B). (3.4)
Writing b A (ε) = -a A (ε) and b B (ε) = -a B (ε), the system can be reduced to

a A (ε) (w A (A) -ln ε) + a B (ε) (w B (A) -ln d ) = u 0 (A), a A (ε) (w A (B) -ln d ) + a B (ε) (w B (B) -ln ε) = u 0 (B). (3.5)
We set

δ(ε) = (w A (B) -ln d )(w B (A) -ln d ) -(w A (A) -ln ε)(w B (B) -ln ε)
and get:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ a A (ε) = -b A (ε) = ln ε -w B (B) δ(ε) u 0 (A) + w B (A) -ln d δ(ε) u 0 (B), a B (ε) = -b B (ε) = w A (B) -ln d δ(ε) u 0 (A) + ln ε -w A (A) δ(ε) u 0 (B).
We check that the leading terms are the same as the ones obtained by considering a single inclusion at A and a single inclusion at B:

a A (ε) = -1 ln ε u 0 (A) + O 1 ln 2 ε and a B (ε) = -1 ln ε u 0 (B) + O 1 ln 2 ε as ε → 0. (3.6)
The presence of two defects appears in a A (ε) and in a B (ε) but only in higher order term in the asymptotic expansion of the coefficients. Setting

w A (x) = ln x -A -w A (x) and w B (x) = ln x -B -w B (x). (3.7)
We follow the algorithmic method of Section 2 and approximate u ε by

u ε = u 0 + a A (ε)w A + a B (ε)w B + r1 ε .
Then, we check that r1

ε satisfies ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -∆r 1 ε = 0 in Ω ε , r1 ε = 0 on ∂Ω 0 , r1 ε = -u 0 -a A (ε)w A -a B (ε)w B on ∂(ω A ε ∪ ω B ε ). For x = A + εX ∈ ∂ω A ε , we have x -B 2 = A -B + εX 2 = d 2 -2ε⟨d, X⟩ + ε 2 , then ln x -B = ln d + k≥1 b k (X)ε k .
Consequently, the remainder r1

ε reads for x = A + εX ∈ ∂ω A ε r1 ε (x) = -u 0 (A) - k≥1 D k u 0 (A)[X, . . . , X] ε k k! -a A (ε) ln ε + ln X -w A (A) - k≥1 D k w A (A)[X, . . . , X] ε k k! -a B (ε) ln d + k≥1 b k (X)ε k -w B (A) - k≥1 D k w B (A)[X, . . . , X] ε k k! .
Using (3.5) and the fact that X ∈ B(0, 1), we check that the terms of low order cancel and thus, for x ∈ ∂ω A ε ,

r1 ε (x) = - k≥1 D k u 0 (A)[X, . . . , X] ε k k! + a A (ε) k≥1 D k w A (A)[X, . . . , X] ε k k! -a B (ε) k≥1 b k (X) - 1 k! D k w B (A)[X, . . . , X] ε k .
We rewrite the remainder on ∂ω A ε as

r1 ε (x) = k≥1 u A k (X)ε k + a A (ε) k≥1 v A k (X)ε k + a B (ε) k≥1 w AB k (X)ε k .
By symmetry, one checks that for x = B + εX ∈ ∂ω B ε , it holds

x -A = B -A + εX = d 2 + 2ε⟨d, X⟩ + ε 2 , then ln x -A = ln d + k≥1 a k (X)ε k , and for x ∈ ∂ω B ε , r1 ε (x) = - k≥1 D k u 0 (B)[X, . . . , X] ε k k! + a B (ε) k≥1 D k w B (B)[X, . . . , X] ε k k! -a A (ε) k≥1 a k (X) - 1 k! D k w A (B)[X, . . . , X] ε k .
We can also rewrite the remainder for

x = B + εX ∈ ∂ω B ε as r1 ε (x) = k≥1 u B k (X)ε k + a B (ε) k≥1 v B k (X)ε k + a A (ε) k≥1 w BA k (X)ε k .
As in the case of one inclusion in Subsection 2.3, we lift each term appearing in the boundary of the inclusions. Thus we approximate u ε by

u ε (x) = u 0 (x) + a A (ε)w A (x) + a B (ε)w B (x) + k≥1 U A k,1 ( x-A ε ) ε k + V A k,1 ( x-A ε ) a A (ε)ε k + W AB k,1 ( x-A ε ) a B (ε)ε k + k≥1 U B k,1 ( x-B ε ) ε k + V B k,1 ( x-B ε ) a B (ε)ε k + W BA k,1 ( x-B ε ) a A (ε)ε k + r2 ε .
These new correctors generate traces on the boundary ∂Ω 0 , exactly as in the case of a single inclusion (see Subsection 2.4): we check that the reminder r2 ε is the solution of

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ -∆r 2 ε = 0 in Ω ε , r2 ε = - k≥1 U A k,1 ( x-A ε ) ε k -V A k,1 ( x-A ε ) a A (ε)ε k + W AB k,1 ( x-A ε ) a B (ε)ε k - k≥1 U B k,1 ( x-B ε ) ε k -V B k,1 ( x-B ε ) a B (ε)ε k + W BA k,1 ( x-B ε ) a A (ε)ε k on ∂Ω 0 , r2 ε = 0 on ∂(ω A ε ∪ ω B ε ). In particular, we get r2 ε H 1 (Ωε) = O(ε 2
) by the argument developed in Remark 2.2 and the order two expansion:

u ε (x) = u 0 (x) + a A (ε)w A (x) + a B (ε)w B (x) + k≥1 U A k,1 ( x-A ε ) ε k + V A k,1 ( x-A ε ) a A (ε)ε k + W AB k,1 ( x-A ε ) a B (ε)ε k + k≥1 U B k,1 ( x-B ε ) ε k + V B k,1 ( x-B ε ) a B (ε)ε k + W BA k,1 ( x-B ε ) a A (ε)ε k + O H 1 (Ωε) (ε 2 ). (3.8)
In the previous expansion many terms are of higher order than the remainder, we have shown:

Theorem 3.1 The solution u ε admits the asymptotic expansion

u ε (x) = u 0 (x) + a A (ε)w A (x) + a B (ε)w B (x) + U A 1,1 ( x-A ε ) ε + V A 1,1 ( x-A ε ) a A (ε)ε + W AB 1,1 ( x-A ε ) a B (ε)ε + U B 1,1 ( x-B ε ) ε + V B 1,1 ( x-B ε ) a B (ε)ε + W BA 1,1 ( x-B ε ) a A (ε)ε + O H 1 (Ωε) (ε 2 ). (3.9)
The algorithm can be continued if one wishes to obtain a higher order expansion. The next step would be to define new correctors in slow variable in order to obtain the following term of the asymptotics:

u ε = u 0 + a A (ε)w A (x) + a B (ε)w B (x) + k≥1 U A k,1 ( x-A ε ) ε k + V A k,1 ( x-A ε ) a A (ε)ε k + W AB k,1 ( x-A ε ) a B (ε)ε k + k≥1 U B k,1 ( x-B ε ) ε k + V B k,1 ( x-B ε ) a B (ε)ε k + W BA k,1 ( x-B ε ) a A (ε)ε k + k≥2 u A k-1,1 (x) ε k + v A k-1,1 (x) a A (ε)ε k + w AB k-1,1 (x) a B (ε)ε k + k≥2 u B k-1,1 (x) ε k + v B k-1,1 (x) a B (ε)ε k + w BA k-1,1 (x) a A (ε)ε k + r3 ε ,
and so on up to the desired precision and then apply the a priori estimates. The study made by V. Maz'ya and S. Poborchi for the case of a single inclusion in [START_REF] Maz ′ Ya | Differentiable functions on bad domains[END_REF]Section 4.1.3] can directly be extended to two separated holes around two fixed points and the estimates (2.14) and (2.15) remains valid. Another interest of this result comes from the theory of topology optimization. This theory initiated in [START_REF] Soko Lowski | On the topological derivative in shape optimization[END_REF][START_REF] Garreau | The topological asymptotic for PDE systems: the elasticity case[END_REF] receives a tremendously growing interest. In a recent paper [START_REF] Amstutz | Optimality conditions for shape and topology optimization subject to a cone constraint[END_REF], S. Amstuzt and M. Ciligot-Travain provide a unifying treatment of constrained shape and topology optimization problems. The variations of shapes using the so-called speed method as well as 'digging holes', the basic concept of sensitivity-based topology optimization, are viewed as a general way to perturb domains. To derive the optimality condition, Amstuzt and Ciligot-Travain make a crucial structure assumption (formulae ( 8) and ( 9) in their work [START_REF] Amstutz | Optimality conditions for shape and topology optimization subject to a cone constraint[END_REF]): an arbitrary number of topological variations should be allowed at the same time, in other words one should be able to justify the leading terms of an asymptotic expansion of a cost function for an arbitrary number of holes dug simultaneously in the domain. However, the large literature on topological optimization deals with a single perturbation. It is then important to check their assumption: for Neumann boundary conditions, it is a consequence of the works of Maz'ya and Mochdan [START_REF] Maz'ya | Uniform asymptotic formulae for Green's functions in singularly perturbed domains[END_REF][START_REF] Maz'ya | Asymptotic treatment of perforated domains without homogenization[END_REF]. Here, we can validate their assumption in the more difficult case of Dirichlet boundary conditions in dimension two. It is a corollary of Theorem 3.1: it suffices to write the leading terms in the expansion (3.9). This term is given by (3.6) and it turns out that it is exactly the superposition of the leading terms of the contribution of each holes considered independently. The interaction appears at higher order in the series in 1 ln ε that defined the fractions a A and a B in ln ε. The passage to an arbitrary number of inclusions is straightforward: the main difference is that the adapted scale h ε has to be computed by solving a larger linear system than (3.4).

Numerical illustration

We use the same function f as in the case of one inclusion. We choose A = (-0.5, 0.5) and B = (0.5, 0.5).

The solution u ε of problem 4.1 with ε = 1 1.5 7 ≃ 0.0585 is computed by a finite element method and given in Figure 8(a). The first remainder r 0 ε = u εu 0 is given in Figure 8(b). The soution w A of problem (3.3) is given in Figure 9(a). The second term of the expansion a A (ε)w A +a B (ε)w B is computed in Figure 9(c). The effect of the two terms approximation is illustrated in Figure 10. Table 2 gives the error norms when we aprroximate u ε by the first terms of its expansion.

u ε -u 0 u ε -u 0 -a A (ε)w A -a B (ε)w B L 2 -

Two moderately close inclusions case

We consider the Dirichlet boundary value problem

-∆u ε = f in Ω ε , u ε = 0 on ∂Ω ε , (4.1)
set in the domain:

Ω ε = Ω 0 ω - ε ∪ ω + ε , with ω ± ε = B(x ± ε , ε) and x ± ε = ±ε α d, (4.2)
as illustrated in Figure 11. We assume that d is a unit vector.

ω - ε ω + ε ε α ε α Ω ε Γ d x - ε • x + ε • 0 • Figure 11:
The perturbed domain

Useful relations

In the construction of the asymptotics expansion, we need to estimate terms where are involved distances between the center of the inclusion x ± ε and a point x on the exterior boundary ∂Ω 0 or on the boundary of the other inclusion ω ∓ ε . This subsection put together these estimates. Let x ∈ ∂Ω 0 , then we can write

x -x ± ε 2 = x 2 1 ∓ 2ε α ⟨x, d⟩ x 2 + ε 2α x 2 , ln x -x ± ε = ln x + k≥1 B± k (x)ε αk , with B± 1 = ∓ ⟨x, d⟩ x 2 . (4.3)
Let x ∈ ∂ω ± ε , there exists X ∈ B(0, 1) such that x = ±(ε α d + εX) and we have

x -x ∓ ε 2 = 4ε 2α 1 + ε 1-α ⟨d, X⟩ + ε 2-2α 4 , ln x -x ∓ ε = ln(2ε α ) + k≥1 ãk (X)ε (1-α)k . (4.4)
Let us make the functions ãk more explicit. We denote h = ε 1-α cos θ + 1 4 ε 2-2α , with θ be such that cos θ = ⟨d, X⟩, then

ln(1 + h) = k≥1 (-1) k+1 k h k = k≥1 (-1) k+1 k k j=0 C k-j k cos k-j θ ε (1-α)(j+k) 4 j = m≥1 ε (1-α)m [ m 2 ]≤k≤m (-1) k+1 (k -1)! (m -k)!(2k -m)! cos 2k-m θ 4 m-k ,
with m = j + k. Using the change of variables p = mk, we have:

m k=[ m 2 ] (-1) k+1 (k -1)! (2k -m)!(m -k)! cos 2k-m θ 2 2m-2k = [ m 2 ] p=0 (-1) m-p+1 (m -p -1)! j!(m -2j)! cos m-2p θ 2 2p .
Using the Tchebychev polynomial functions (see [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]Chap. 22]), we have m 2

[ m 2 ] p=0 (-1) p (m -p -1)! j!(m -2p)! 2 m-2p cos m-2p θ = cos(mθ).

First term in slow variable

As in the cases of one inclusion or two separated inclusions, we look for coefficients a ± (ε) and b(ε) such that the remainder r 1 ε defined by

u ε (x) = u 0 (x) + a + (ε) ln x -x + ε + a -(ε) ln x -x - ε + b(ε) w(x) + r 1 ε (x),
with w defined in (2.5), is reduced with respect to r 0

ε = u ε -u 0 . The remainder r 1 ε satisfies ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -∆r 1 ε = 0 in Ω ε , r 1 ε (x) = -a + (ε) ln x -x + ε -a -(ε) ln x -x - ε -b(ε) ln x on ∂Ω 0 , r 1 ε (x) = -u 0 (x) -a + (ε) ln x -x + ε -a -(ε) ln x -x - ε -b(ε)w(x) on ∂ω ± ε .
For x ∈ ∂Ω 0 , we have

r 1 ε (x) = O(1) ⇔ a + (ε) + a -(ε) + b(ε) = O(1).
For x = ε α d + εX ∈ ∂ω + ε , with X ∈ B(0, 1), we have

r 1 ε (x) = -u 0 (0) + O(ε α ) -a + (ε) ln εX -a -(ε) ln 2ε α d + εX -b(ε)(w(0) + O(ε α )).
Consequently

r 1 ε (x) = O(1) ⇔ u 0 (0) + a + (ε) ln ε + a -(ε) ln 2ε α + w(0) b(ε) = O(1).
We obtain the similar condition on ∂ω - ε :

r 1 ε (x) = O(1) ⇔ u 0 (0) + a + (ε) ln 2ε α + a -(ε) ln ε + w(0) b(ε) = O(1).
Hence we solve the linear system in the unknowns (a

± (ε), b(ε)): ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ a + (ε) + a -(ε) + b(ε) = 0, ln ε a + (ε) + (ln 2 + α ln ε) a -(ε) + w(0) b(ε) = -u 0 (0), (ln 2 + α ln ε) a + (ε) + ln ε a -(ε) + w(0) b(ε) = -u 0 (0), to set a + (ε) = a -(ε) = u 0 (0) 2w(0) -ln 2 -(1 + α) ln ε and b(ε) = -2u 0 (0) 2w(0) -ln 2 -(1 + α) ln ε .
As in (2.6) for the case of one inclusion, a new scale appears:

h ε = 1 2w(0) -ln 2 -(1 + α) ln ε = -1 (1 + α) ln ε + O 1 ln 2 ε as ε → 0. ( 4.7)
A notable point is that the coefficient α measuring the distance between the two inclusion appears as a multiplicative factor in the leading term of the asymptotic. The situations of sections 3 and 4 are drastically different. We introduce the normalized corrector w ε as

w ε (x) = ln x -x + ε + ln x -x - ε -2w(x), (4.8)
with w defined by (2.5). Then we approximate u ε by

u ε (x) = u 0 (x) + u 0 (0)w ε (x) h ε + r1 ε (x).

Second terms

We check that the remainder r1

ε satisfies ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -∆r 1 ε = 0 in Ω ε , r1 ε = -u 0 (0)w ε h ε on ∂Ω 0 , r1 ε = -u 0 -u 0 (0)w ε h ε on ∂ω ± ε .
In order to estimate the first reminder r1 ε , we examine its traces on Ω ε , that is on ∂Ω 0 and on ∂ω ± ε . For this, we use relations established in Subsection 4.1. For any x ∈ ∂Ω 0 , we can write

r1 ε (x) = k≥2 (B + k (x) + B - k (x)) h ε ε αk , ∀x ∈ ∂Ω 0 , (4.9)
with B ± k = -u 0 (0) B± k and B± k defined in (4.3). We have

r1 ε H 1 2 (∂Ω 0 ) = O ε 2α ln ε .
Let us now look at the trace on the inclusions ω ± ε . Let x ∈ ∂ω ± ε , and X ∈ B(0, 1) be such that x = ±(ε α d + εX). We have

ln x -x ± ε = ln ε. Then r1 ε (x) = -u 0 (x) -u 0 (0) ln ε + ln x -x ∓ ε -2w(x) h ε . Using Subsection 4.1, we can write relation (4.4) ln x -x ∓ ε = ln(2ε α ) + k≥1 ãk (X)ε (1-α)k ,
with ãk defined in (4.5) and satisfying (4.6). By interior elliptic regularity, u 0 and w are analytic in the vicinity of 0. Then

u 0 (x) = u 0 (0) + n≥1 D n u 0 (0)[x, . . . , x] = u 0 (0) + j+k≥1 (±1) j+k C j j+k D j+k u 0 (0)[X, . . . , X j times , d, . . . , d k times ]ε j+αk , w(x) = w(0) + n≥1 D n w(0)[x, . . . , x],
in the vicinity of 0. For the convenience, we denote

a k (X) = -u 0 (0)ã k (X), u ± j,k (X) = (±1) j+k+1 C j j+k D j+k u 0 (0)[X j , d k ], w ± j,k (X) = (±1) j+k 2u 0 (0)C j j+k D j+k w(0)[X j , d k ],
with a k defined in (4.5). Then the remainder r1 ε reads

r1 ε (x) = j+k>0 u ± j,k (X) ε j+αk + k≥1 a k (X) h ε ε (1-α)k + j+k>0 w ± j,k (X) h ε ε j+αk , ∀x ∈ ∂ω ± ε .
(4.10) As for the case of one inclusion, the mean value property insures that ∂B(0,1) u ± j,k = 0 and ∂B(0,1)

w ± j,k = 0. Now we need to lift each boundary condition appearing in (4.9) and (4.10).

• The functions B ± k introduced in (4.3) to estimate the remainder on the exterior boundary as given in (4.9) generate correctors b ± k defined by

-∆b ± k = 0 in Ω 0 , b ± k = B ± k on ∂Ω 0 . (4.11)
These correctors do not satisfy the Dirichlet condition on the boundary of the inclusion ∂ω ± ε and then generate errors on these boundaries.

• We now proceed as in the case of one inclusion when we lift u k and w k . We notice that the mean value property is satisfied since ∂B(0,1)

a k = 0, ∂B (0,1)
u ± j,k = 0 and ∂B(0,1) w ± j,k = 0.

Then Lemma 2.1 (point 3) can be applied. We set A k , U ± j,k and V ± j,k the solution of problem (2.4)

with v = a k , v = u ± j,k and v = w ± j,k respectively: -∆A k = 0 in R 2 ω, A k = a k on ∂ω, -∆U ± j,k = 0 in R 2 ω, U ± j,k = u ± j,k on ∂ω, -∆W ± j,k = 0 in R 2 ω, W ± j,k = w ± j,k on ∂ω.
These functions have similar decomposition as in (2.8) and (2.9).

We then approximate u ε by

u ε (x) = u 0 (x) + u 0 (0)w ε (x) h ε + k≥2 (b + k (x) + b - k (x)) h ε ε αk + k≥1 A k ( x-x + ε ε ) + A k ( x-x - ε ε ) h ε ε (1-α)k + j+k>0 U + j,k ( x-x + ε ε ) + U - j,k ( x-x - ε ε ) ε j+αk + j+k>0 W + j,k ( x-x + ε ε ) + W - j,k ( x-x - ε ε ) h ε ε j+αk + r2 ε (x). (4.12)

Third terms

Let us study now the remainder r2

ε . It satisfies ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ -∆r 2 ε = 0 in Ω ε , r2 ε (x) = - k≥1 A k ( x-x + ε ε ) + A k ( x-x - ε ε ) h ε ε (1-α)k - j+k>0 U + j,k ( x-x + ε ε ) + U - j,k ( x-x - ε ε ) ε j+αk - j+k>0 W + j,k ( x-x + ε ε ) + W - j,k ( x-x - ε ε ) h ε ε j+αk for x ∈ ∂Ω 0 , r2 ε = - k≥2 (b + k + b - k ) h ε ε αk on ∂ω ± ε .
• Let us study the trace on ∂Ω 0 . According to their decomposition like in (2.8) and (2.9), the traces T A ± k , T U j,k and T W j,k on ∂Ω 0 of the functions A k (

x-x ± ε ε ), U ± j,k ( x-x ± ε ε ) and W ± j,k ( x-x ± ε ε ) satisfy T A ± k H 1 2 (∂Ω 0 ) = O(ε), T U ± j,k H 1 2 (∂Ω 0 ) = O(ε) and T W ± j,k H 1 2 (∂Ω 0 ) = O(ε).
We now lift these functions in slow variables and define â± k , û± j,k and ŵ± j,k as the solution of

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -∆v = 0 in Ω 0 , v = - 1 ε T V on ∂Ω 0 ,
with T V = T A k (

x-x ± ε ε ), T U ± j,k (

x-x ± ε ε ) and T V ± j,k (

x-x ± ε ε ) respectively.

• To lift the boundary condition on ∂ω ± ε , we proceed as in the first step of the construction of the asymptotic expansion (see also the study of the remainder r3 ε in the case of one inclusion). The function b ± k have a value at 0 which is a priori not 0. We can lift the boundary condition by considering the function -

k≥2 (b + k (0) + b - k (0))w ε (x) h 2 ε ε αk .
We can read u ε :

u ε (x) = u 0 (x) + u 0 (0)w ε (x) h ε + k≥2 (b + k (x) + b - k (x)) h ε ε αk - k≥2 (b + k (0) + b - k (0))w ε (x) h 2 ε ε αk + k≥1 A k ( x-x + ε ε ) + A k ( x-x - ε ε ) h ε ε (1-α)k + k≥2 (â + k-1 (x) + â- k-1 (x)) h ε ε k-α(k-1) + j+k>0 U + j,k ( x-x + ε ε ) + U - j,k ( x-x - ε ε ) ε j+αk + j+k>1,j≥1 (û + j-1,k (x) + û- j-1,k (x)) ε j+αk + j+k>0 W + j,k ( x-x + ε ε ) + W - j,k ( x-x - ε ε ) h ε ε j+αk + j+k>1,j≥1
( ŵ+ j-1,k (x) + ŵj-1,k (x)) h ε ε j+αk + r3 ε (x).

Antsatz

We inject the remainder r3 ε in the equation and obtain

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ -∆r 3 ε = 0 in Ω ε , r3 ε = k≥2 (b + k (0) + b - k (0))w ε h 2 ε ε αk on ∂Ω 0 , r3 ε = - k≥2 (â + k-1 (x) + â- k-1 (x)) h ε ε k-α(k-1) - j+k>1,j≥2 (û + j-1,k + û- j-1,k ) ε j+αk - j+k>1,j≥2
( ŵ+ j-1,k + ŵj-1,k ) h ε ε j+αk on ∂ω ± ε .

We have now to reiterate the previous step with the new functions. This justify the building of the antsatz u ε (x) = u 0 (x) + u 0 (0)w ε (x) h ε + i≥0,j≥1,k≥0 c i,j,k w ε (x)h i ε ε j+αk + i≥1,k≥2

β i,k (b + k (x) + b - k (x)) h i ε ε αk + i≥1,k≥2 β i,k (b + k (0) + b - k (0))w ε (x) h i+1 ε ε αk + i≥0,j≥0,k≥1 (A i,j,k ( x-x + ε ε ) + A i,j,k ( x-x - ε ε )) h i ε ε j+k(1-α) + i≥0,j≥0,k≥1
âi,j,k (x) h i ε ε j+1+k(1-α) + i≥0,j+k>0

(V + i,j,k ( x-x + ε ε ) + V - i,j,k ( x-x - ε ε ))h i ε ε j+αk + i≥0,j+k>0
(v + i,j,k (x) + vi,j,k (x))h i ε ε j+1+αk .

To justify it, we proceed exactly as in Section 2.6.

Numerical illustration

We use the same test as in the case of one inclusion. The solution u ε of problem 4.1 with ε = 1 1.5 7 ≃ 0.0585 and α = 0.8 is computed by a finite element method and given in Figure 12(a). With these parameters, we have

h ε ≃ -0.

Figure 1 : u 0

 10 Figure 1: u 0

Figure 3 :Figure 4 :

 34 Figure 3: First order approximation u εu 0

Figure 5 :

 5 Figure 5: Second order approximation u 0 + u 0 (0)w h ε

  (a) u0 -u0(0)w ln ε (b) uε -u0 + u0(0)w ln ε

Figure 6 :

 6 Figure 6: Second order approximation u 0u 0 (0)w ln ε u εu 0 u εu 0u 0 (0)w h ε u εu 0 + u 0 (0)w ln ε L 2 -norm 7.799 e-2 6.88 e-3 6.56 e-3 H 1 -norm 4.0277 e-1 2.865 e-2 5.290 e-2

  (a) uε (b) r 0 ε = uε -u0

Figure 8 :

 8 Figure 8: First term approximation

Figure 9 :Figure 10 :

 910 Figure 9: Second term of the expansion

067 and - 1 ( 1 +

 11 α) ln ε ≃ 0.049. The first remainder r 0 ε = u ε -u 0 is given in Figure 12(b). The second term of the expansion

  (a) uε (b) r 0 ε = uε -u0

Figure 12 :

 12 Figure 12: First term approximation

  .

Figure 13 :Figure 14 :Figure 15 :

 131415 Figure 13: Second term of the expansion

Table 1 :

 1 Error norms 3 Two separated inclusions case

	norm 7.799 e-2	6.88 e-3	6.56 e-3
	H 1 -norm 4.0277 e-1	2.865 e-2	5.290 e-2

Now fix two distinct points A et B in Ω and set d = → AB. We consider the Dirichlet boundary value problem:

Table 2 :

 2 Error norms

	norm 8.323 e-2	7.76 e-3
	H 1 -norm 4.2756 e-1	4.946 e-2

Table 3 :

 3 Error norms