ERROR ANALYSIS OF A BACKWARD EULER POSITIVE PRESERVING STABILIZED SCHEME FOR A CHEMOTAXIS SYSTEM P. CHATZIPANTELIDIS AND C. PERVOLIANAKIS

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ERROR ANALYSIS OF A BACKWARD EULER POSITIVE PRESERVING STABILIZED

SCHEME FOR A CHEMOTAXIS SYSTEM

P. CHATZIPANTELIDIS AND C. PERVOLIANAKIS

Abstract. For a Keller-Segel model for chemotaxis in two spatial dimensions we consider a modiﬁcation of a

positivity preserving fully discrete scheme using a local extremum diminishing ﬂux limiter. We discretize space

using piecewise linear ﬁnite elements on an quasiuniform triangulation of acute type and time by the backward

Euler method. We assume that initial data are suﬃciently small in order not to have a blow-up of the solution.

Under appropriate assumptions on the regularity of the exact solution and the time step parameter we show

existence of the fully discrete approximation and derive error bounds in L2for the cell density and H1for the

chemical concentration. We also present numerical experiments to illustrate the theoretical results.

1. Introduction

We shall consider a Keller-Segel system of equations of parabolic-parabolic type, where we seek u=u(x, t)

and c=c(x, t) for (x, t)∈Ω×[0, T ],satisfying

(1.1) 









ut= ∆u−λdiv (u∇c),in Ω ×[0, T ],

ct= ∆c−c+u, in Ω ×[0, T ],

∂u

∂ν = 0,∂c

∂ν = 0,on ∂Ω×[0, T ],

u(·,0) = u0, c(·,0) = c0,in Ω,

where Ω ⊂R2is a convex bounded domain with boundary ∂Ω, νis the outer unit normal vector to ∂Ω, ∂/∂ν

denotes diﬀerentiation along νon ∂Ω, λis a positive constant and u0, c0≥0, u0̸= 0.

The chemotaxis model (1.1) describes the aggregation of slime molds resulting from their chemotactic features,

cf. e.g. [18]. The function uis the cell density of cellular slime molds, cis the concentration of the chemical

substance secreted by molds themselves, c−uis the ratio of generation or extinction, and λis a chemotactic

sensitivity constant.

There exists an extensive mathematical study of chemotaxis models, cf. e.g., [24,15,16,17,29] and references

therein. It is well-known that the solution of (1.1) may blow up in ﬁnite time. However, if ∥u0∥L1(Ω) ≤4πλ−1,

the solution (u, c) of (1.1) exists for all time and is bounded in L∞, cf. e.g. [23].

A key feature of the system (1.1) is the conservation of the solution uin L1norm,

∥u(t)∥L1(Ω) =∥u0∥L1(Ω),0≤t≤T,(1.2)

which is an immediate result of the preservation of non-negativity of u,

u0≥0,̸≡ 0 in Ω ⇒u > 0 in Ω ×(0, T ],(1.3)

and the conservation of total mass ZΩ

u(x, t)dx =ZΩ

u0(x)dx, 0≤t≤T.(1.4)

Capturing blowing up solutions numerically is a challenging problem and many numerical methods have been

proposed to address this. The main diﬃculty in constructing suitable numerical schemes is to preserve several

essential properties of the Keller-Segel equations such as positivity, mass conservation, and energy dissipation.

Some numerical schemes were developed with positive-preserving conditions, cf. e.g. [14,9,8], which depend

on a particular spatial discretization and impose CFL restrictions on the time step. Other approaches include,

ﬁnite-volume based numerical methods, [9,8], high-order discontinuous Galerkin methods, [12,13,20], a ﬂux

corrected ﬁnite element method, [28], and a novel numerical method based on symmetric reformulation of the

chemotaxis system, [21]. For a more detailed review on recent developments of numerical methods for chemotaxis

problems, we refer to [10,30].

In order to maintain the total mass and the non-negativity of the numerical approximations of the system

(1.1), Saito in [26] proposed and analyzed a fully discrete method that uses an upwind ﬁnite element scheme in

space and backward Euler method in time. His proposed ﬁnite element scheme made use of Baba and Tabata’s

upwind approximation, see [1]. Strehl et al. in [28] proposed a slightly diﬀerent approach. The stabilization was

implemented at a pure algebraic level via algebraic ﬂux correction, see [19]. This stabilization technique can be

Date: today is July 24, 2024.

2020 Mathematics Subject Classiﬁcation. 65M60, 65M15.

Key words and phrases. ﬁnite element method, error analysis, nonlinear parabolic problem, chemotaxis, positivity preservation.

arXiv:2210.04709v5 [math.NA] 23 Jul 2024

2 P. CHATZIPANTELIDIS AND C. PERVOLIANAKIS

applied to high order ﬁnite element methods and maintain the mass conservation and the non-negativity of the

solution.

In the variational form of (1.1), we seek u(·, t)∈H1and c(·, t)∈H1,for t∈[0, T ], such that

(1.5) (ut, v)+(∇u−λu∇c, ∇v) = 0,∀v∈H1,with u(0) = u0,

(ct, v)+(∇c, ∇v)+(c−u, v) = 0,∀v∈H1,with c(0) = c0,

where (f, g) = RΩfg dx.

In our analysis we consider regular triangulations Th={K}of Ω, with h= maxK∈ThhK,hK= diam(K),

and the ﬁnite element spaces

(1.6) Sh:= χ∈ C0:χ|K∈P1,∀K∈ Th,

where C0=C0(Ω) denotes the continuous functions on Ω.

A semidiscrete approximation of the variational problem (1.5) is: Find uh(t)∈ Shand ch(t)∈ Sh, for t∈[0, T ],

such that

(1.7) (uh,t, χ)+(∇uh−λuh∇ch,∇χ)=0,∀χ∈ Sh,with uh(0) = u0

(ch,t, χ)+(∇ch,∇χ)+(ch−uh, χ)=0,∀χ∈ Sh,with ch(0) = c0

where u0

h, c0

h∈ Sh.

We now formulate (1.7) in matrix form. Let Zh={Zj}N

j=1 be the set of nodes in Thand {ϕj}N

j=1 ⊂ Shthe

corresponding nodal basis, with ϕj(Zi) = δij .Then, we may write uh(t) = PN

j=1 αj(t)ϕj, with u0

h=PN

j=1 α0

jϕj

and ch(t) = PN

j=1 βj(t)ϕj, with c0

h=PN

j=1 β0

jϕj. Thus, the semidiscrete problem (1.7) may then be expressed,

with α= (α1, . . . , αN)Tand β= (β1, . . . , βN)T, as

Mα′(t)+(S−Tβ)α(t)=0,for t∈[0, T ],with α(0) = α0,(1.8)

Mβ′(t)+(S+M)β(t) = Mα(t),for t∈[0, T ],with β(0) = β0,(1.9)

where α0= (α0

1, . . . , α0

N)T,β0= (β0

1, . . . , β0

N)T,M= (mij ), mij = (ϕi, ϕj), S= (sij ), sij = (∇ϕi,∇ϕj),

Tβ= (τij (β)) and τij (β) = λ

ℓ=1

βℓ(ϕj∇ϕℓ,∇ϕi),for i, j = 1, . . . , N.

We will often suppress the index βin the coeﬃcients τij =τij (β) and in T=Tβ. The matrices Mand Sare

both symmetric and positive deﬁnite, however Tdue to the chemotactical ﬂux λu(t)∇c(t) is not symmetric.

Note that the semidiscrete solutions uh(t), ch(t) of (1.7) are nonnegative if and only if the coeﬃcient vectors

α(t),β(t) are nonnegative elementwise. In order to ensure nonnegativity, we may employ the lumped mass

method, which results from replacing the mass matrix Min (1.8)-(1.9) with a diagonal matrix MLwith elements

j=1 mij .

A suﬃcient condition for α(t) to be nonnegative elementwise is that the oﬀ diagonal elements of S−Tare

nonpositive. Further, for β(t) to be nonnegative elementwise, it suﬃces that the oﬀ diagonal elements of Sare

nonpositive.

Assuming, that Thsatisﬁes an acute condition, i.e., all interior angles of a triangle K∈ Thare less than π/2, we

have that sij ≤0, cf. e.g., [11]. Then, in order to ensure that the oﬀ diagonal elements of S−Tare nonpositive

we may add an artiﬁcial diﬀusion operator D=Dβ. This technique is commonly used in conversation laws,

cf. e.g. [19] and references therein. This modiﬁcation of the semidiscrete scheme (1.5) is proposed in [28]. This

scheme is often called low-order scheme since we introduce an error which manifests in the order of convergence.

To improve the convergence order of the low-order scheme, Strehl et al. in [28] proposed another scheme,

which is called algebraic ﬂux correction scheme or AFC scheme. To derive the AFC scheme we decompose the

error, introduced in the low-order scheme by adding the artiﬁcial diﬀusion operator, into internodal ﬂuxes. Then

we appropriately restore high accuracy in regions where the solution does not violate the non-negativity. There

exists various algorithms to limit the internodal ﬂuxes. We will consider limiters that will satisfy the discrete

maximum principle and linearity preservation on arbitary meshes, as the one proposed by G. Barrenechea et al.

in [4].

Our purpose here, is to analyze fully discrete schemes, for the approximation of (1.1), by discretizing in time

the low-order scheme and the AFC scheme using the backward Euler method. We will consider the case where

the solution of (1.1) remains bounded for all t≥0, therefore we will assume that ∥u0∥L1(Ω) ≤4λ−1π.

Our analysis of the stabilized schemes is based on the corresponding one employed by Barrenechea et al. in

[3]. In order to show existence of the solutions of the nonlinear fully discrete schemes, we employ a ﬁxed point

argument and demonstrate that our approximations remain uniformly bounded, provided that This quasiuniform

and our time step parameter k, is such that k=O(h1+ϵ), with 0 <ϵ<1.

ERROR ANALYSIS OF POSITIVE PRESERVING SCHEMES FOR CHEMOTAXIS 3

A B

αβ

Zi8

Zi7

Zi6

Zi5

Zi4

Zi3

Zi2

Zi1

Figure 2.1. Various sub-domains of the triangulation Th.

We shall use standard notation for the Lebesgue and Sobolev spaces, namely we denote Wm

p=Wm

p(Ω),

Hm=Wm

2,Lp=Lp(Ω), and with ∥·∥m,p =∥·∥Wm

p,∥·∥m=∥·∥Hm,∥·∥Lp=∥·∥Lp(Ω), for m∈Nand

p∈[1,∞], the corresponding norms.

The fully discrete schemes we consider approximate (un, cn) by (Un, Cn)∈ Sh× Shwhere un=u(·, tn),

cn=c(·, tn), tn=nk,n= 0, . . . , NTand NT∈N,NT≥1, k=T/NT. Assuming that the solutions (u, c) of

(1.1) are suﬃciently smooth, with u∈W2

p, with p∈(2,∞] and c∈W2,∞, we derive error estimates of the form

∥Un−un∥L2≤C(k+k−1/2h2+h3/2|log h|),

∥Cn−cn∥1≤C(k+k−1/2h2+h).

The paper is organized as follows: In Section 2we introduce notation and the semidiscrete low-order scheme

and the AFC scheme for the discretization of (1.1). Further, we prove some auxiliary results for the stabilization

terms, that we will employ in the analysis that follows and rewrite the low-order and AFC scheme, as general

semidiscrete scheme. In Section 3, we discretize the general semidiscrete scheme in time, using the backward

Euler method. For a suﬃciently smooth solution of (1.1) and k=O(h1+ϵ), with 0 <ϵ<1, we demonstrate that

there exists a unique discrete solution which remains bounded and derive error estimates in L2for the cell density

and H1for the chemical concentration. In Section 4, we show that the discrete solution preserves nonnegativity.

Finally, in Section 5, we present numerical experiments, illustrating our theoretical results.

2. Preliminaries

2.1. Mesh assumptions. We consider a family of regular triangulations Th={K}of a convex polygonal

domain Ω ⊂R2. We will assume that the family Thsatisﬁes the following assumption.

Assumption 2.1. Let Th={K}be a family of regular triangulations of Ωsuch that any edge of any Kis either

a subset of the boundary ∂Ωor an edge of another K∈ Th, and in addition

(1) This shape regular, i.e, there exists a constant γ > 0,independent of Kand Th,such that

(2.1) hK

ϱK≤γ, ∀K∈ Th,

where ϱK=diam(BK), and BKis the inscribed ball in K.

(2) The family of triangulations This quasiuniform, i.e., there exists constant ϱ > 0such that

maxK∈ThhK

minK∈ThhK≤ϱ, ∀K∈ Th,(2.2)

(3) All interior angles of K∈ Thare less than π/2.

Let Nh:= {i:Zia node of the triangulation Th},Ehbe the set of all edges of the triangulation Thand eij ∈ Eh

denotes an edge of Thwith endpoints Zi,Zj∈ Zh. We denote ωethe collection of triangles with a common edge

e∈ Eh, see Fig. 2.1, and ωi,i∈ Nh, the collection of triangles with a common vertex Zi, i.e. ωi=∪Zi∈KK,

see Fig. 2.1. The sets Zh(ω) and Eh(ω) contain the vertices and the edges, respectively, of a subset of ω⊂ Th

and Zi

h:= {j:Zj∈ Zh,adjacent to Zi}. Using the fact that This shape regular, there exists a constant κγ,

independent of h, such that the number of vertices in Zi

his less than κγ, for i= 1, . . . , N.

Since Thsatisﬁes (2.2), we have for all χ∈ Sh,cf., e.g., [6, Chapter 4],

(2.3) ∥χ∥L∞+∥∇χ∥L2≤Ch−1∥χ∥L2and ∥∇χ∥L∞≤Ch−1∥∇χ∥L2.

Further, in our analysis, we will employ the following trace inequality which holds for K∈ Th, cf. e.g., [6,

Theorem 1.6.6] together with a homogeneity argument, cf. e.g., [25, Theorem 3.1.2],

(2.4) ∥v∥L1(∂K)≤Cγ∥v∥H1(K),

where Cγa positive constant that depends on γ.

2.2. Stabilized semidiscrete methods. In the sequel we will present two stabilized semidiscrete schemes for

the numerical approximation of (1.1), namely the low order scheme and the AFC scheme, which have been

proposed in [28].

4 P. CHATZIPANTELIDIS AND C. PERVOLIANAKIS

2.2.1. Low order scheme. The semidiscrete problem (1.7) may be expressed in the matrix form (1.8)–(1.9),

where the matrices Mand Sare symmetric and positive deﬁnite. However Tβis not symmetric, but as we will

demonstrate in the sequel, cf. Lemma 3.4, it has a zero-column sum.

For a function v∈ C0, let v= (v1, . . . , vN)Tdenote the vector with coeﬃcients its nodal values vi=v(Zi),

Zi∈ Zh,i= 1, . . . , N. We will often expess the coeﬃcients τij ,i, j = 1, . . . , N, of Tβas functions of an element

ψ∈ Sh,τij =τij (ψ) = τij (ψ), such that ψ=Pjψjϕj∈ Shand ψ= (ψ1, . . . , ψN)T. Thus the elements of

Tβ=T= (τij ), may be expressed equivallently as,

(2.5) τij =τij (ψ) = τij (ψ) = λ(ϕj∇ψ, ∇ϕi) = λ

ℓ=1

ψℓ(ϕj∇ϕℓ,∇ϕi).

In order to preserve non-negativity of α(t) and β(t), a low order semi-discrete scheme of minimal model has

been proposed, cf. e.g. [28], where Mis replaced by the corresponding lumped mass matrix MLand an artiﬁcial

artiﬁcial diﬀusion operator D=Dβ= (dij ) is added to T, to elliminate all negative oﬀ-diagonal elements of T,

so that T+D≥0, elementwise. Thus, assuming that Thsatisﬁes an acute condition, i.e., all interior angles of

a triangle K∈ Thare less than π/2, gives that sij ≤0 and hence, the oﬀ diagonal elements of S−T−Dare

nonpositive, sij −τij −dij ≤0, i̸=j,i, j = 1, . . . , N. However, note that assuming Thto be acute is not a

necessary condition to preserve non-negativity of α(t) or β(t).

Also, we will often suppress the index βin the coeﬃcients dij =dij (β), i, j = 1, . . . , N , or expess them as

functions of an element ψ∈ Sh,dij (ψ) = dij (ψ), such that ψ=Pjψjϕj∈ Shand ψ= (ψ1, . . . , ψN)T. Since,

we would like our scheme to maintain the mass, Dmust be symmetric with zero row and column sums, cf. [28],

which is true if D= (dij )N

i,j=1 is deﬁned by

(2.6) dij := max{−τij ,0,−τji}=dji ≥0,∀j̸=iand dii := −X

j̸=i

dij .

Thus the resulting system for the approximation of (1.1) is expressed as follows, we seek α(t),β(t)∈RNsuch

that, for t∈[0, T ],

MLα′(t)+(S−Tβ−Dβ)α(t)=0,with α(0) = α0,(2.7)

MLβ′(t)+(S+ML)β(t) = MLα(t),with β(0) = β0.(2.8)

Let for w∈ Sh,dh(w;·,·) : C0× C0→R,be a bilinear form deﬁned by

(2.9) dh(w;v, z) :=

i,j=1

dij (w)(vi−vj)zi,∀v, z ∈ C0,

and (·,·)hbe an inner product in Shthat approximates (·,·) and is deﬁned by

(2.10) (ψ, χ)h=X

K∈Th

h(ψχ),with QK

h(g) = 1

3|K|X

Z∈Zh(K)

g(Z)≈ZK

g(x)dx,

with Zh(K) the vertices of K∈ Thand |K|the area of K∈ Th. Then following [3], the coupled system (2.7)–(2.8)

can be rewritten in the following variational formulation: Find uh(t), ch(t)∈ Shsuch that

(uh,t, χ)h+ (∇uh−λuh∇ch,∇χ) + dh(ch;uh, χ) = 0,∀χ∈ Sh,(2.11)

(ch,t, χ)h+ (∇ch,∇χ)+(ch−uh, χ)h= 0,∀χ∈ Sh,(2.12)

with uh(0) = u0

h∈ Shand ch(0) = c0

h∈ Sh.

We can easily see that (·,·)hinduces an equivallent norm to ∥·∥L2on Sh. Thus, there exist constants C, C′

independed on h, such that

(2.13) C∥χ∥h≤ ∥χ∥L2≤C′∥χ∥h,with ∥χ∥h= (χ, χ)1/2

h,∀χ∈ Sh.

2.2.2. Algebraic ﬂux correction scheme. The replacement of the standard FEM discretization (1.8)-(1.9) by the

low-order scheme (2.7)-(2.8) ensures nonnegativity but introduces an error which manifests the order of conver-

gence, cf. e.g. [27,19]. Thus, following Strehl et al. [27], one may “correct” the semidiscrete scheme (2.7)-(2.8)

by introducing a ﬂux correction term. Hence, we also consider an algebraic ﬂux correction (AFC) scheme, which

involves the decomposition of this error into internodal ﬂuxes, which can be used to restore high accuracy in

regions where the solution is well resolved and no modiﬁcations of the standard FEM are required. There exists

various algorithms to implement an AFC scheme. Here we will follow the one proposed by G. Barrenachea et.

al. in [4].

The AFC scheme is constructed in the following way. Let f= (f1, . . . , fN)Tdenote the error of inserting the

operator Dβin (1.8), i.e., f(α,β) = −Dβα.Using the zero row sum property of matrix Dβ, cf. (2.6), we can

ERROR ANALYSIS OF POSITIVE PRESERVING SCHEMES FOR CHEMOTAXIS 5

show that the residual admits a conservative decomposition into internodal ﬂuxes,

(2.14) fi=X

j̸=i

fij , fji =−fij , i = 1, . . . , N,

where the amount of mass transported by the raw antidiﬀusive ﬂux fij is given by

(2.15) fij := fij (α,β)=(αi−αj)dij (β),∀j̸=i, i, j = 1, . . . , N.

For the rest of this paper we will call the internodal ﬂuxes as anti-diﬀusive ﬂuxes. Some of these anti-diﬀusive

ﬂuxes are harmless but others may be responsible for the violation of non-negativity. Such ﬂuxes need to be

canceled or limited so as to keep the scheme non-negative. Thus, every anti-diﬀusive ﬂux fij is multiplied by a

solution-depended correction factor aij ∈[0,1], to be deﬁned in the sequel, before it is inserted into the equation.

Hence, the AFC scheme is the following: We seek α(t),β(t)∈RNsuch that, for t∈[0, T ],

MLα′(t)+(S−Tβ−Dβ)α(t) = f(α(t),β(t)),with α(0) = α0,(2.16)

MLβ′(t)+(S+ML)β(t) = MLα(t),with β(0) = β0,(2.17)

where f(α(t),β(t)) = (f1, . . . , fN)T, with

(2.18) fi:= fi(α(t),β(t)) = X

j̸=i

aij fij , i = 1, . . . , N,

and aij ∈[0,1] are appropriately deﬁned in view of the antidiﬀusive ﬂuxes fij .

In order to determine the coeﬃcients aij , one has to ﬁx a set of nonnegative coeﬃcients qi. In principle the

choice of these parameters qican be arbitrary. But eﬃciency and accuracy can dictate a strategy, which does

not depend on the ﬂuxes fij but on the type of problem ones tries to solve and the mesh parameters. We will

not ellaborate more on the choice of qi, for a more detail presentation we refer to [19] and [3] and the references

therein. In the sequel we will employ two particular choices of qicf. Lemma 2.7.

To ensure that the AFC scheme maintains the nonnegativity property, it is suﬃcient to choose the correction

factors aij such that the sum of anti-diﬀusive ﬂuxes is constrained, (cf. e.g., [19]), as follows. Let qi>0, i∈ Nh,

are given constants that do not depend on αand

(2.19) Q+

i=qi(αmax

i−αi) and Q−

i=qi(αmin

i−αi), i ∈ Nh,

with αmax

i,αmin

ithe local maximum and local minimum at ωi. Then aij should satisfy

(2.20) Q−

i≤X

j̸=i

aij fij ≤Q+

i, i ∈ Nh.

Remark 2.1. Note that if all the correction factors aij = 0, then the AFC scheme (2.16)–(2.17)reduces to the

low-order scheme (2.7)–(2.8).

Remark 2.2. The criterion (2.20)by which the correction factors are chosen, implies that the limiters used in

(2.16)–(2.17)guarantee that the scheme is non-negative. In fact, if αiis a local maximum in ωi, then (2.20)

implies the cancellation of all positive ﬂuxes. Similarly, all negative ﬂuxes are canceled if αiis a local minimum

in ωi. In other words, a local maximum cannot increase and a local minimum cannot decrease. As a consequence,

aij fij cannot create an undershoot or overshoot at node i.

We shall compute the correction factors aij using Algorithm 2.3, which has originally proposed by Kuzmin,

cf. [19, Section 4]. Then we have that

Q−

i≤R−

iP−

i≤X

j̸=i

aij fij ≤R+

iP+

i≤Q+

i, i ∈ Nh,

which implies that (2.20) holds.

Algorithm 2.3 (Computation of correction factors aij ).Given data:

(1) The positive coeﬃcients qi, such that qi=O(h),i∈ Nh.

(2) The ﬂuxes fij ,i̸=j,i, j = 1, . . . , N.

(3) The coeﬃcients αj, βj,j= 1, . . . , N.

Computation of factors aij .

(1) Compute the limited sums P±

i:= P±

i(α,β),i∈ Nh, of positive and negative anti-diﬀusive ﬂuxes

P+

i=X

j̸=i

max{0, fij }and P−

i=X

j̸=i

min{0, fij }, i ∈ Nh.

(2) Retrieve the local extremum diminishing upper and lower bounds Q±

i:= Q±

i(α), i ∈ Nh,

Q+

i=qi(αmax

i−αi),and Q−

i=qi(αmin

i−αi), i ∈ Nh,

where αmax

i, αmin

iare the local maximum and local minimum in ωi.

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ERRORANALYSISOFABACKWARDEULERPOSITIVEPRESERVINGSTABILIZEDSCHEMEFORACHEMOTAXISSYSTEMP.CHATZIPANTELIDISANDC.PERVOLIANAKISAbstract.ForaKeller-Segelmodelforchemotaxisintwospatialdimensionsweconsideramodificationofapositivitypreservingfullydiscreteschemeusingalocalextremumdiminishingfluxlimiter.Wediscret...

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ERROR ANALYSIS OF A BACKWARD EULER POSITIVE PRESERVING STABILIZED SCHEME FOR A CHEMOTAXIS SYSTEM P. CHATZIPANTELIDIS AND C. PERVOLIANAKIS

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