Cristóbal Bertoglio

Carlos Conca

David Nolte

Grigory Panasenko

Konstantinas

Konstantin Pileckas

Junction of models of different dimension for flows in tube structures by Womersley-type interface conditions

Keywords: Stokes and Navier-Stokes equations, thin structures, asymptotic partial decomposition, hybrid dimension models AMS classification: 35Q35, 76D07, 65N55

come

Introduction

The Stokes and Navier-Stokes equations in thin tube structures are the most classical models for a viscous flow in pipelines or blood vessels. Tube structures are domains which are tree-like sets of thin cylinders (or thin rectangles in two-dimensional setting). The ratio of the diameters of cylinders to their heights (or ratio of the sides of rectangles) is a small parameter ε. The method of asymptotic partial decomposition of a domain (MAPDD) allows to reduce essentially the computer resources needed for the numerical solution of such problems. This method combines the full-dimensional description in some neighborhoods of bifurcations and a reduced-dimensional description out of these small subdomains and it prescribes some special junction conditions at the interface between these 3D and 1D submodels (see [START_REF] Blanc | Asymptotic analysis and partial asymptotic decomposition of the domain for Stokes equation in tube structure[END_REF][START_REF] Panasenko | Method of asymptotic partial decomposition of domain[END_REF][START_REF] Panasenko | Partial asymptotic decomposition of domain: Navier-Stokes equation in tube structure[END_REF][START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF]). In particular, for the non-steady Navier-Stokes equations these interface conditions prescribe a pre-computed Poiseuille type shape. To this end one has to solve a Jordanian chain of elliptic equations on the section and take their linear combination [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF]. This condition is justified for the Navier-Stokes equation without a boundary layer in time, when the data of the problem have vanishing all derivatives in time at the initial moment. However in the case of a general setting (see [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure[END_REF]) the question on the high order interface conditions is still open. The goal of the paper is to give and justify a more general interface condition which is applicable for the problems with a boundary layer in time. Such condition is constructed for the steady state Stokes equations and then is generalized for the non-stationary Navier-Stokes equations.

In this new version the trial and test functions have vanishing transversal components of the velocity and vanishing normal derivative of the normal component inside the cylinders, instead of the pre-computed Poiseuille-type shape. This also leads to a easy-to-implement finite element formulation of the MAPDD and to assess it numerically in dependance of the Reynolds number.

The remainder of paper is organized as follows. In Section 2 the full-dimensional Dirichlet's problem for the non-stationary Navier-Stokes equations and stationary Stokes equations in a thin tube structure are formulated. We give two weak formulations: one containing only the unknown velocity (formulation "without pressure" which is convenient for the asymptotic analysis) and one formulation containing both unknown velocity and unknown pressure which is convenient for the numerical solution. In Section 3 the original MAPDD method is revisited. In Section 4 the new version of MAPDD for the steady Stokes and transient Navier-Stokes equations is introduced and the main theorems summarized. For the sake of readability by a wide range of specialists the proofs are moved to the Appendices. Finally, we present some numerical examples in Section 5 where the theoretical results are confirmed.

Note that recently an asymptotic analysis of flows of complex rheology in thin tube structures was developed in [START_REF] Bunoiu | Asymptotic analysis of a Bingham fluid in a thin T-like shaped structure[END_REF], [START_REF] Benes | Effective flow of incompressible micropolar fluid through a system of thin pipes[END_REF].

Thin tube structure domain

Let us remind the definition of a thin tube structure [START_REF] Panasenko | Asymptotic expansion of the solution of Navier-Stokes equation in a tube structure[END_REF][START_REF] Panasenko | Multi-scale Modeling for Structures and Composites[END_REF][START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF], and graphically exemplified in Figure 1. Let O 1 , O 2 , . . . , O N be N different points in R n , n = 2, 3, and e 1 , e 2 , . . . , e M be M closed segments each connecting two of these points (i.e. each e j = O i j O k j , where i j , k j ∈ {1, . . . , N }, i j = k j ).

Γ in ω 1 ε S 12 B ε,δ 1 B dec,ε 1,2 δ S 21 Γ out ω 2 ε B ε,δ
All points O i are supposed to be the ends of some segments e j . The segments e j are called edges of the graph. The points O i are called nodes. Any two edges e j and e i , i = j, can intersect only at the common node. A node is called vertex if it is an end point of only one edge. Assume that the set of vertices is O N 1 +1 , O N 1 +2 , . . . , O N , where N 1 < N .

Denote B =

M j=1 e j the union of edges and assume that B is a connected set. The graph G is defined as the collection of nodes and edges.

Let e be some edge, e = O i O j . Consider two Cartesian coordinate systems in R n . The first one has the origin in O i and the axis O i x (e) 1 has the direction of the ray [O i O j ); the second one has the origin in O j and the opposite direction, i.e. O j

x(e) 1 is directed over the ray [O j O i ). With every edge e j we associate a bounded domain σ j ⊂ R n-1 having a C 2 -smooth boundary ∂σ j , j = 1, . . . , M . For every edge e j = e and associated σ j = σ (e) we denote by B (e) ε the cylinder

B (e) ε = {x (e) ∈ R n : x (e)
1 ∈ (0, |e|),

x (e) ε ∈ σ (e) },
where x (e) = (x (e) 2 , . . . , x

(e)
n ), |e| is the length of the edge e and ε > 0 is a small parameter. Notice that the edges e j and Cartesian coordinates of nodes and vertices O j , as well as the domains σ j , do not depend on ε.

Denoting σ (e) ε = {x (e) ∈ R n-1 :

x (e) ε ∈ σ (e) } we can write B (e)

ε = (0, |e|) × σ (e)
ε . Let ω 1 , . . . , ω N be bounded independent of ε domains in R n with Lipschitz boundaries ∂ω j ; introduce the nodal domains

ω j ε = {x ∈ R n : x -O j ε ∈ ω j }. Denote d = max 1≤j≤N diamω j
. By a tube structure we call the following domain

B ε = M j=1 B (e j ) ε N j=1 ω j ε .
So, the tube structure B ε is a union of all thin cylinders having edges as the heights plus small smoothing domains ω j ε in the neighborhoods of the nodes. Their role is to avoid artificial corners in the boundary of intersecting cylinders. Furthermore, we will assume that B ε is a bounded domain (connected open set) with a C 2 -smooth boundary.

The full dimension fluid flow problem

Through the paper we will consider the stationary Stokes or the non-stationary Navier-Stokes equations in B ε with the no-slip conditions at the boundary ∂B ε except for some parts γ j ε of the boundary where the velocity field is given as known inflows and outflows (for alternative boundary conditions on the inlet and outlet boundaries of the domain, the reader is referred to [START_REF] Bègue | Going back to Stokes and Naiver-Stokes equations with boundary conditions on the pressure[END_REF][START_REF] Bertoglio | Benchmark problems for numerical treatment of backflow at open boundaries[END_REF]).

Let us define these parts of the boundary. Denote

γ j ε = ∂ω j ε ∩ ∂B ε , γ j = ∂ω j ∩ ∂B j 1 where B j 1 = {y : yε + O j ∈ B ε } and γ ε = ∪ N j=N 1 +1 γ j ε .
Let us introduce first the initial boundary value problem for the non-stationary Navier-Stokes equations.

∂uε ∂t -ν∆u ε + (u ε • ∇)u ε + ∇p ε = 0, divu ε = 0, u ε ∂Bε = g ε , u ε (x, 0) = 0, (1)
where u ε is the unknown velocity vector, p ε is the unknown pressure, g ε is a given vector-valued function satisfying the following conditions:

g ε (x, t) = g j ( x -O j ε , t) if x ∈ γ j ε , j = N 1 + 1, .
.., N and equal to zero for the remaining part of the boundary ∂B ε \γ ε . Here g j :

γ j × [0, +∞) → R n belonging to C [ J+4 2 ]+1 ([0, T ]; H 3/2 0 (γ j ))
, T is a positive number. Assume that g j | t=0 = 0 and (the compatibility condition)

∂Bε g ε • nds = 0. (2)
i.e.

N j=N 1 +1 γ j ε g j ( x -O j ε , t) • nds = 0.
Remark 1.In this case one can prove that g ε has a divergence free extension g defined in

B ε × [0, T ] which we denote by the same symbol g ε , g ε ∈ C [ J+4 2 ]+1 ([0, T ]; H 2 (B ε )
) satisfying for all t ∈ [0, T ] the following asymptotic estimates

g ε L 2 (Bε) + g ε,t L 2 (Bε) + g ε,tt L 2 (Bε) ≤ cε n-1 2 ; ∇g ε L 2 (Bε) + ∇g ε,t L 2 (Bε) ≤ cε n-3 2 , ∆g ε L 2 (Bε) ≤ cε n-5 2 , n = 2, 3, (3)
where the constant c is independent of ε (See Lemma A1 in Appendix).

There are two equivalent weak formulations of the problem: "with pressure" and "without pressure" which differ by the space of test functions; in the formulation "without pressure" test functions are divergence free and so the integral containing the pressure disappears; the only unknown function is the vector of velocity. In the formulation "with pressure" the space of test functions is wider, they may not be divergence free, so that the pressure parcipates in the formulation as an unknown function. The formulation "without pressure" is used mainly in analysis while the definition "with pressure" is more convenient for the numerical approximation using finite elements because it doesn't require construction of divergence free bases in the space of test functions.

Introduce the space H 1 div0(∂Bε\γε) (B ε ) as the subspace of vector valued functions from H 1 (B ε ) satisfying the conditions div v = 0, v| ∂Bε\γε = 0, i.e.,

H 1 div0(∂Bε\γε) (B ε ) = v ∈ H 1 (B ε )| div v = 0; v| ∂Bε\γε = 0 .
We consider as well the smaller subspace By a weak solution we understand the couple of the vector-field u ε and a scalar function

H 1 div0 (B ε ) = H 1 div0(∂Bε\γε) (B ε ) ∩ H 1 0 (B ε ) of
p ε such that u ε (x, 0) = 0, u ε ∈ L 2 (0, T ; H 1 div0(∂Bε\γε) (B ε )), u εt ∈ L 2 (0, T ; L 2 (B ε )), p ε ∈ L 2 (0, T ; L 2 (B ε )), u ε = g ε on γ ε and (u ε , p ε ) satisfy the integral identity for every vector-field φ ∈ H 1 0 (B ε ) for all t ∈ (0, T ), Bε u εt • φ + ν∇u ε : ∇φ + (u ε , ∇u ε ) • φ dx = Bε p ε divφdx. ( 4
)
Replacing the space of test functions by a subspace of divergence free functions we get another weak formulation without the integral Bε p ε divφdx.

Definition 1.2. By a weak solution we understand the vector-field u ε such that

u ε (x, 0) = 0, u ε ∈ L 2 (0, T ; H 1 div0(∂Bε\γε) (B ε )), u εt ∈ L 2 (0, T ; L 2 (B ε )), u ε = g ε on γ ε and u ε satisfies the integral identity for every vector-field φ ∈ H 1 div0 (B ε ) for all t ∈ (0, T ), Bε u εt • φ + ν∇u ε : ∇φ + (u ε , ∇u ε ) • φ dx = 0. ( 5
)
For sufficiently small ε there exists a unique solution to this problem (see [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF]). The equivalence of these formulations follows from [START_REF] Ladyzhenskaya | The Mathematical Theory of Viscous Incompressible Fluid[END_REF], see also [START_REF] Temam | Navier-Stokes equations; theory and numerical analysis[END_REF].

Consider the Dirichlet's boundary value problem for the stationary Stokes equation:

             -ν∆u ε + ∇p ε = 0, x ∈ B ε , divu ε = 0, x ∈ B ε , u ε = g ε , x ∈ ∂(B ε ) , (6)
ν is a positive constant, g ε is a given vector-valued function satisfying the following conditions:

g ε (x) = g j ( x -O j ε ) if x ∈ γ j ε , j = N 1 + 1, .
.., N (O j are vertices!)and equal to zero for the remaining part of the boundary ∂B ε \γ ε . Here g j : γ j → R n belonging to H 3/2 0 (γ j ). Assume that (the compatibility condition)

∂Bε g ε • nds = 0. (7)
i.e.

N j=N 1 +1 γ j ε g j ( x -O j ε ) • nds = 0.
Remark 2.In the stationary case as well one can prove that g ε has a divergence free extension g defined in B ε which we denote by the same symbol g ε ,

g ε ∈ H 2 (B ε )) (see Lemma A1 in the Appendix).
Let as give two equivalent definitions of a weak solution. First one is "with pressure". Definition 1.1'. By a weak solution we understand the couple of the vector-field u ε and a scalar function

p ε such that u ε ∈ H 1 div0(∂Bε\γε) (B ε ), p ε ∈ L 2 (B ε ), u ε = g ε on γ ε and (u ε , p ε ) satisfy the integral identity: for any test function v ∈ H 1 0 (B ε ) ν Bε ∇u ε (x) : ∇v(x)dx = Bε p ε divφdx. . ( 8
)
The second is "without pressure". Definition 1.2'. By a weak solution we understand the vector-field u ε such that u ε ∈ H 1 div0(∂Bε\γε) (B ε ), u ε = g ε on γ ε and u ε satisfies the integral identity: for any test function

v ∈ H 1 div0 (B ε ) ν Bε ∇u ε (x) : ∇v(x)dx = 0. . ( 9
)
It is well known that there exists a unique solution to this problem (see [START_REF] Ladyzhenskaya | The Mathematical Theory of Viscous Incompressible Fluid[END_REF]). The equivalence of these formulations follows from [START_REF] Ladyzhenskaya | The Mathematical Theory of Viscous Incompressible Fluid[END_REF], see also [START_REF] Temam | Navier-Stokes equations; theory and numerical analysis[END_REF]. -ν∆v P (y) = 1 , y ∈ σ (e) , v P (y) = 0 , y ∈ ∂σ (e) .

MAPDD: the classical version

(

) 10
If e has the cosines directors k e1 , ..., k en and the local variables x (e) are related to the global ones by equation x (e) = x (e) (x) then the Poiseuille flow is

V (e) P (x) = const (k e1 v P ((x (e) (x)) /ε), ..., k en v P ((x (e) (x)) /ε)) T , x = (x 2 , ..., x n ).
In the case const = 1 denote the Poiseuille flow V 0,(e) P ; it is the normalized Poiseuille flow.

Let δ be a small positive number much greater than ε but much smaller than 1. For any edge e = O i O j of the graph introduce two hyperplanes orthogonal to this edge and crossing it at the distance δ from its ends, see Figure 1.

Denote the cross-sections of the cylinder B (e) ε by these two hyperplanes respectively, by S i,j (the cross-section at the distance δ from O i ), and S j,i (the cross-section at the distance δ from O j ), and denote the part of the cylinder between these two cross-sections by B dec,ε ij . Denote B ε,δ i the connected truncated by the cross sections S i,j , part of B ε containing the vertex or the node O i .

Define the subspace H 1,δ div0 (B ε ) (and respectively,

H 1,δ div0(∂Bε\γε) (B ε )) of the space H 1 div0 (B ε ) (re- spectively of H 1 div0(∂Bε\γε) (B ε )
) such that on every truncated cylinder B dec,ε ij its elements described in local variables x (e) for the edge e (vector-valued functions) have a form of the Poiseuille flow V (e) P (x).

The MAPDD replaces the original full-dimensional problem for the steady Stokes equations (6) by the following weak formulation:

Find

u ε,δ ∈ H 1,δ div0(∂Bε\γε) (B ε ) such that u ε,δ = g ε on γ ε and for all v ∈ H 1,δ div0 (B ε ), ν Bε ∇u ε,δ (x) : ∇v(x)dx = 0 . ( 11
)
For the non-stationary Navier-Stokes equations the Poiseuille flow has more complicated structure [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF]. For small ε it can be approximated by a time dependent linear combination of vectorvalued functions V P,1 (x), ..., V P,J (x) such that in local variables their first component v P,j (y) satisfies a Jordanian chain of equations -ν∆v P,j+1 (y) = -v P,j (y) , y ∈ σ (e) , v P,j+1 (y) = 0 , y ∈ ∂σ (e) . [START_REF] Ladyzhenskaya | Boundary Value Problems of Mathematical Physics[END_REF] while the transversal components of vectors V P,1 (x), ..., V P,J (x) are equal to zero, V P,1 (x) = V P (x) (the steady Poiseuille flow), and so the space of test functions for the MAPDD H 1,δ div0 (B ε ) is a subspace of H 1 div0 (B ε ) such that on every truncated cylinder B dec,ε ij its elements described in local variables x (e) for the edge e (vector-valued functions) have a form of linear combinations of these functions α 1 V P,1 (x) + ... + α J V P,J (x), α 1 , ..., α J are real numbers.

Define as well the space H 1,δ div0(∂Bε\γε) (B ε ) as a similar subspace of H 1 div0(∂Bε\γε) (B ε ). The weak formulation of the classical version of MAPDD for the non-stationary Navier-Stokes problem is given in [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF]. It is equivalent to the following formulation without pressure:

By a weak solution we understand the vector-field u ε,δ such that

u ε,δ (x, 0) = 0, u ε,δ ∈ L ∞ (0, T ; H 1,δ div0(∂Bε\γε) (B ε )), u ε,δ,t ∈ L 2 (0, T ; L 2 (B ε )),
u ε,δ = g ε on γ ε and u ε,δ satisfies the integral identity for every vector-field

φ ∈ H 1,δ div0 (B ε ) for all t ∈ (0, T ), Bε u ε,δ,t • φ + ν∇u ε,δ : ∇φ + (u ε,δ , ∇u ε,δ ) • φ dx = 0. ( 13
)
Existence and uniqueness of a solution for sufficiently small ε is proved as in [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF] by Galerkin method.

Summary of main results on the classical version

For the classical version of MAPDD the theorem on the error estimates is proved. Namely, it was proved that given J there exists a constant C independent of ε such that if δ = CJε| ln(ε)| then for the Stokes equations the following estimate holds [START_REF] Panasenko | Partial asymptotic decomposition of domain: Navier-Stokes equation in tube structure[END_REF], [START_REF] Blanc | Asymptotic analysis and partial asymptotic decomposition of the domain for Stokes equation in tube structure[END_REF]:

u ε -u ε,δ H 1 (Bε) = O(ε J ). ( 14
)
For the non-stationary Navier-Stokes equations we have the following result [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF]:

Given natural number J, if g j ∈ C [ J+4 2 ]+1 ([0, T ]; W 3/2,2 (∂ω j )
) and there exists an interval (0, τ ), τ > 0 such that g j = 0 for t ∈ (0, τ ) then there exists a constant C (independent of ε and J) such that if δ = CJε| ln ε| then

sup t∈(0,T ) u ε,δ -u ε L 2 (Bε) + ∇u ε,δ -u ε L 2 ((0,T );L 2 (Bε)) = O(ε J ) . ( 15
)
Although this classical version of the MAPDD is an effective method reducing considerably the computational costs it does not work in the situation when the above condition g j = 0 for t ∈ (0, τ ) is not satisfied. Indeed, in [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure[END_REF] it was shown that for small values of time of order ε 2 linear combinations of functions V P,i are no more a good approximations for the velocity inside the tubes, they should be replaced by the "boundary layer-in-time". Moreover, the coordinate change from velocity degrees-of-freedom to α 1 , . . . , α J involves intrusive modifications of the numerical simulation software, for both system assembly and linear algebra parts.

MAPDD: the new junction conditions

We now propose a new, more general, formulation of the method involving new junction conditions. The advantages are twofold: (1) it removes the condition g j = 0 for t ∈ (0, τ ), therefore being applicable for arbitrary transient regimes, and (2) it considerable simplifies the numerical implementation in the context of finite elements since only additional, easy-to-build integral terms need to be added to a standard weak form.

Formulation of the new version

Let us define the subspace

H 1,δ div0 (B ε ) (and respectively, H 1,δ div0(∂Bε\γε) (B ε )) of the space H 1 div0 (B ε ) (respectively of H 1 div0(∂Bε\γε) (B ε
) ) in a different way, so that on every truncated cylinder B dec,ε ij its elements described in local variables (vector-valued functions) have vanishing trasversal (tangential) components while the longitudinal (normal) component has vanishing longitudinal (normal) derivative. Namely, if the local variables x (e) for the edge e coinside with the global ones x then they have a form of the Womersley flow

W (e) P (x) = (v 1 (x /ε), 0, ..., 0) T . v 1 ∈ H 1 0 (σ (e) )
. If e has the cosines directors k e1 , ..., k en and the local variables x (e) are related to the global ones by equation x (e) = x (e) (x) then they are

W (e) P (x) = const (k e1 v 1 ((x (e) (x)) /ε), ..., k en v 1 ((x (e) (x)) /ε)) T , x = (x 2 , ..., x n ).
As in the classical version the method of asymptotic partial domain decomposition (MAPDD) replaces the problem (6) by its projection on this newly defined space H 1,δ div0(∂Bε\γε) (B ε ). Note that this space is wider than the space of test functions in the classical version because the steady Poiseuille flow is a particular case of functions W (e) P . The weak formulations repeat literally the formulations of the previous section but with respect to the newly defined space H 1,δ div0(∂Bε\γε) (B ε ).

Stokes equations

Consider the Stokes equations [START_REF] Blanc | Asymptotic analysis and partial asymptotic decomposition of the domain for Stokes equation in tube structure[END_REF].

The new version of the method of asymptotic partial domain decomposition (MAPDD) replaces the problem ( 6) by its projection on

H 1,δ div0(∂Bε\γε) (B ε ): to find u ε,δ ∈ H 1,δ div0(∂Bε\γε) (B ε ), such that u ε,δ = g ε on γ ε and satisfies the following integral identityfor all vector fields ∀v ∈ H 1,δ div0 (B ε ) ν Bε ∇u ε,δ (x) : ∇v(x)dx = 0. ( 16
)
Applying the Lax-Milgram argument one can prove that there exists a unique solution u ε,δ of the partially decomposed problem.

Remark 3.The classical version of MAPDD differs from this new one by the definition of the space on which we project the problem. Namely, in the new version the projection is taken onto the

space H 1,δ div0(∂Bε\γε) (B ε ) involving the Womersley functions, while in the classical case [START_REF] Panasenko | Multi-scale Modeling for Structures and Composites[END_REF] it is a subspace of H 1 div0(∂Bε\γε) (B ε ) such that on every truncated cylinder B dec,ε ij its elements are equal to a Poiseuille flow V (e) P . Note that the space Womersley functions is much wider than the space of Poiseuille flows.

Estimate for the difference between the exact solution and the MAPDD solution: asymptotic analysis of the Stokes equations

Theorem 1. Given natural number J there exists a constant C (independent of ε and

J) such that if δ = CJε| ln ε|, then u ε -u ε,δ H 1 (Bε) = O(ε J ) . ( 17
)
Proof. This estimate is the same as in the classical version of the MAPDD. The proof is similar to that of the classical version. However, for the sake of completeness we give it in the Appendix.

Navier-Stokes equations

Consider the Navier-Stokes equations [START_REF] Alnaes | The FEniCS Project Version 1.5[END_REF].

The new version of the method of asymptotic partial domain decomposition (MAPDD) replaces the problem (1) by [START_REF] Ladyzhenskaya | On some problems of vector analysis and generalized formulations of boundary value problems for the Navier-Stokes equations[END_REF], where the space H 1,δ div0(∂Bε\γε) (B ε ) is replaced by the newly defined space of divergence free vector-functions having the Womersley form within cylinders B dec,ε ij : by a weak solution we understand the vector-field

u ε,δ such that u ε,δ (x, 0) = 0, u ε,δ ∈ L ∞ (0, T ; H 1,δ div0(∂Bε\γε) (B ε )), u ε,δ,t ∈ L 2 (0, T ; L 2 (B ε )), u ε,δ = g ε on γ ε and u ε,δ satisfies the integral identity for every vector-field φ ∈ H 1,δ div0 (B ε ) for all t ∈ (0, T ), Bε u ε,δ,t • φ + ν∇u ε,δ : ∇φ + (u ε,δ , ∇u ε,δ ) • φ dx = 0. ( 18
)
The existence and uniqueness of its solution is proved as in [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF]. Let us give the formulation "with pressure". Note that it is less evident than for the full dimension problem. First note that knowing the velocity field u ε,δ , solution to problem [START_REF] Panasenko | Partial asymptotic decomposition of domain: Navier-Stokes equation in tube structure[END_REF], we can reconstitute some function p ε,δ which is interpreted as the MAPDD pressure. Namely, let us denote U ij (x (e) , t) the trace of the solution u ε,δ to problem (18) at every cross-section S ij . Then we get a standard Navier-Stokes problem in each domain B ε,δ i with the known boundary value

U ij (x (e) , t) on S ij , the no-slip boundary condition on ∂B ε,δ i \Σ i if i = 1, ..., N 1 , or on ∂B ε,δ i \(Σ i ∪γ ε i ) if i = N 1 + 1, ..

., N and respectively with condition

U ij = g ε at γ ε i in the last case; the initial condition is U ij (x, 0) = 0. Here Σ i is a union ∪ j:O i O j ∈{e 1 ,...,e M } S ij of all cross-sections S ij belonging to the boundary of B ε,δ i
. This problem admits a unique solution-velocity (coinciding with u ε,δ ) and a pressure p ε,δ,i unique up to an additive function θ i of t. Let us introduce an extended space of the test functions

H 1,δ 0 (B ε ) = {φ ∈ H 1 0 (B ε )|φ(x) = W (e) P (x), x ∈ B dec,ε ij , e = O i O j ; ∂B ε,δ i φ • n = 0, i = 1, ..., N }
and extend the integral identity [START_REF] Panasenko | Partial asymptotic decomposition of domain: Navier-Stokes equation in tube structure[END_REF] for test functions of this space:

Bε u ε,δ,t • φ + ν∇u ε,δ : ∇φ + (u ε,δ , ∇u ε,δ ) • φ dx = = N i=1 B ε,δ i u ε,δ,t • φ -ν∆u ε,δ • φ + (u ε,δ , ∇u ε,δ ) • φ dx + j:O i O j ∈{e 1 ,...,e M } ∂B ε,δ i ∩S ij ν ∂u ε,δ ∂n φds + + M l=1 d l σ (e l ) ε u ε,δ,t • φ + ν∇ x (e l ), u ε,δ : ∇ x (e l ), φdx (e l ), = = N i=1 - B ε,δ i ∇p ε,δ,i • φdx + j:O i O j ∈{e 1 ,...,e M } ∂B ε,δ i ∩S ij ν ∂u ε,δ ∂n • φds + + M l=1 d l σ (e l ) ε u ε,δ,t • φ + ν∇ x (e l ), u ε,δ : ∇ x (e l ), φdx (e l ), = = N i=1 B ε,δ i p ε,δ,i divφdx + j:O i O j ∈{e 1 ,...,e M } ∂B ε,δ i ∩S ij ν ∂u ε,δ ∂n -p ε,δ n • φds + + M l=1 d l σ (e l ) ε u ε,δ,t • φ + ν∇ x (e l ), u ε,δ : ∇ x (e l ), φdx (e l ), ,
where for e l = O i O j , d l is the distance between the cross sections S ij and S ji , n is an outer normal vector for B ε,δ i . Using the condition

∂B ε,δ i φ • n = 0, i = 1, ..., N (19)
we will prove in Appendix that the sum of the last two sums of integrals is equal to zero. So, the variational formulation is: find the vector-field u ε,δ and the pressure p ε,δ such that

u ε,δ (x, 0) = 0, u ε,δ ∈ L ∞ (0, T ; H 1,δ div0(∂Bε\γε) (B ε )), u ε,δ,t ∈ L 2 (0, T ; L 2 (B ε )), u ε,δ = g ε on γ ε ,p ε,δ ∈ L 2 (0, T ; L 2 (B ε,δ i )
) for all i = 1, ..., N , and the couple (u ε,δ , p ε,δ ) satisfies the integral identity for every vector-field φ ∈ H 1,δ 0 (B ε ) for all t ∈ (0, T ):

Bε u ε,δ,t • φ + ν∇u ε,δ : ∇φ + u ε,δ , ∇u ε,δ ) • φ dx = N i=1 B ε,δ i p ε,δ divφdx (20)
Note that so defined pressure is not unique, it is defined up to function

θ i (t) in each subdomain B ε,δ i , i = 1, ..., N .
A similar weak formulation with pressure can be given for the Stokes equations. Note that if N = M + 1 (number of nodes and vertices is equal to the number of edges plus one) then the restriction [START_REF] Panasenko | Multi-scale Modeling for Structures and Composites[END_REF] can be removed from the definition of space H 1,δ 0 (B ε ) and then the number of undetermined constants θ i (t) in the variational formulation [START_REF] Panasenko | Flows in a tube structure: equation on the graph[END_REF] will be reduced to one function θ N (t), so that the pressure in the reduced geometry is defined up to a constant as in the case of full geometry. This assertion will be proved in the Appendix.

The numerical tests are held for such geometries with N = M + 1. In this case it is possible to apply the restriction divu ε,δ = 0 on the solution directly in equation ( 20) so that a considerably simpler-to-implement formulation holds true: find the vector-field u ε,δ and the pressure p

ε,δ such that u ε,δ (x, 0) = 0, u ε,δ ∈ L ∞ (0, T ; H 1 (B ε,δ i )), for all i = 1, ..., N , u ε,δ,t ∈ L 2 (0, T ; L 2 (B ε,δ i )), u ε,δ = g ε at γ ε , u ε,δ = 0 at (∂B ε,δ i ∩ ∂B ε )\γ ε , p ε,δ ∈ L 2 (0, T ; L 2 (B ε,δ i )) for all i = 1, ..., N , u ε,δ • t = 0 on S ij ∪ S ji , u ε,δ • n S ij + u ε,δ • n S ji = 0.
where t is the unit tangent vector, and the couple (u ε,δ , p ε,δ ) satisfies for all t ∈ (0, T ) the integral identity for every vector-

field φ ∈ H 1 (B ε,δ i ), q ∈ L 2 (B ε,δ i ), for all i = 1, ..., N , such that φ = 0 at ∂B ε,δ i ∩ ∂B ε ,

and for all edges

O i O j , φ • t = 0 at S ij ∪ S ji and φ • n| S ij + φ • n| S ji = 0 : N i=1 B ε,δ i u ε,δ,t • φ + ν∇u ε,δ : ∇φ + (u ε,δ , ∇u ε,δ ) • φ -p ε,δ divφ + qdivu ε,δ dx + M l=1 d l σ (e l ) ε u ε,δ,t • φ + ν∇ x (e l ), u ε,δ : ∇ x (e l ), φdx (e l ), = 0. ( 21
)
Finally, note that the last two terms in ( 21) are analogous to the ones obtained in the context the so called Stokes-consistent methods for backflow stabilization at open boundaries [START_REF] Bertoglio | A Stokes-residual backflow stabilization method applied to physiological flows[END_REF].

Estimate for the difference between the exact solution and the MAPDD solution for the non-stationary Navier-Stokes equations

The result of the previous section can be generalized for the non-stationary problem for the Navier-Stokes equations (1) using the approach of [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF] and [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure[END_REF]. Assume that g ε introduced in the second section has a divergence free extension gε defined in

B ε with g ∈ C [ J+4 2 ]+1 ([0, T ]; H 2 (B ε )
) satisfying for all t ∈ [0, T ] the following asymptotic estimates

g L 2 (Bε) + gt L 2 (Bε) + gtt L 2 (Bε) ≤ cε n-1 2 ; ∇g L 2 (Bε) + ∇g t L 2 (Bε) ≤ cε n-3 2 , ∆g L 2 (Bε) ≤ cε n-5 2 , n = 2, 3, (22)
where the constant c is independent of ε.

As before,

∂Bε g ε • nds = 0. ( 23
) Theorem 2.Let g j ∈ C [ J+4 2 ]+1 ([0, T ]; W 3/2,2 (∂ω j ))
. Given natural number J there exists a constant C (independent of ε and J) such that if δ = CJε| ln ε|, then

sup t∈(0,T ) u ε,δ -u ε L 2 (Bε) + ∇(u ε,δ -u ε ) L 2 ((0,T );L 2 (Bε)) = O(ε J ) . ( 24
)

Numerical examples

In this section, the previous analysis is complemented by numerical experiments for the new MAPDD formulation applied to the stationary Stokes problem and the transient Navier-Stokes problem, for a sequence of values of ε.

The errors of the MAPDD solutions obtained in the truncated domain with respect to reference solutions computed in the full domain are evaluated in the norms given by Eqs. ( 17), [START_REF] Pileckas | Three-dimensional solenoidal vectors[END_REF].

Problem setup

Consider the two-dimensional geometry illustrated in Fig. 1. Two junctions are connected by a straight tube. This straight tube (labeled B dec,ε 1,2 ) is included in the full reference model, or truncated when the reduced MAPDD model is used.

The radius of the tube is proportional to ε (we set R = ε). For each value of ε, the junction domains are contracted homothetically by a factor of ε with respect to the center points marked with plus signs in Fig. 1. The distance between these points, L, remains the same for all values of ε. Straight tube extensions (blue areas, B ε,δ 1;2 ) are added to the junction domains. Theorem 1 requires the associated distance, δ, from the centers of the junction domains to the interfaces, to be 2) is added for convenience, to cancel with the ln(ε) terms and leave rational numbers as the interface coordinates.

δ = Cε| ln(ε)|. ( 25

Stationary Stokes test case

Since one of our main motivations is the numerical simulation of blood flows, we choose for the viscosity and the density values that represent physiologically relevant conditions, assuming the fluid is incompressible and Newtonian. Typical parameters of blood are a dynamic viscosity of µ = 0.035cm 2 /s and a density of ρ = 1g/cm 3 . Remind the relation between the dynamic viscosity µ and the kinematic viscosity ν: ν = µ/ρ. At the inlet Γ in of the upstream junction domain a Dirichlet boundary condition for the velocity is defined as

g ε = 0, 1.5U 0 (1 -(x 1 -c 0 ) 2 /ε 2 ) T ,
where c 0 is the x 1 coordinate of the center of the boundary and U 0 is chosen such that Re = 2ρεU 0 /µ = 1. A homogeneous Neumann boundary condition for the normal stress is applied on the outlet Γ out of the downstream junction domain.

Transient Navier-Stokes test case

In the transient Navier-Stokes test case, the physical constants are set to the same values as for the Stokes problem, i.e., µ = 0.035 cm 2 /s and ρ = 1 g/cm 3 .

A pulsating inflow velocity is defined on Γ in via Dirichlet boundary conditions as

g ε = 0, 1.5U 0 (1 -(x 1 -c 0 ) 2 /ε 2 ) sin(πt/T ) T ,
where t is the actual time and T = 0.8 s is the duration of a cycle. U 0 is computed from the Reynolds number, Re = 2ρεU 0 /µ. As for the Stokes problem, a homogeneous Neumann boundary condition defines the outflow on Γ out .

For the convergence study, Reynolds numbers Re = 1, 25, 50, 80 and 100 are considered. In addition, we analyze the MAPDD model for a high Reynolds number of Re = 2500.

Numerical discretization

A mixed finite element method is adopted for discretizing the Stokes and Navier-Stokes equations. We use monolithic velocity-pressure coupling with inf-sup stable second order Taylor-Hood elements on unstructured, uniform triangle meshes. The transient Navier-Stokes problem is discretized in time with the implicit Euler method. The convection term, written in skew-symmetric form, is treated semi-implicitly. The time step size is ∆t = 0.01 s. The time interval of the simulations is a half cycle, i.e., 0 ≤ t ≤ T /2.

The numerical meshes of the domains are created such that the number of elements along the tube diameter is approximately 20 for each value of ε. The average grid size at the boundaries is therefore h = ε/10. This results in 170592 elements in the full domain for the smallest value of ε = 2 -6 and C = 2/ ln(2), which corresponds to 784037 degrees of freedom in the Navier-Stokes system. The triangulation of the corresponding reduced domain consists of 15366 elements and the solution space contains 70741 degrees of freedoms.

The problem is implemented and solved using the FEniCS finite element library [START_REF] Alnaes | The FEniCS Project Version 1.5[END_REF]. The numerical meshes are generated with Gmsh [START_REF] Geuzaine | Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities[END_REF].

Results

Stationary Stokes test case

The velocity and pressure field of the stationary Stokes problem, computed with the full model and with the MAPDD method, are illustrated in Fig. 2 for the largest value of ε = 0.5. No visible differences exist between the full and the MAPDD results.

The velocity error of the MAPDD model with respect to the full reference solution is analyzed quantitatively in Fig. 3 for the full range of values of ε. The error is computed in the H 1 (B ε ) norm, cf. [START_REF] Panasenko | Asymptotic expansion of the solution of Navier-Stokes equation in a tube structure[END_REF] in Theorem 1. Note that the error estimate depends on the solutions in the full domain, B ε . The mesh nodes of the MAPDD and the full domains match for the junctions. In the truncated tube, the MAPDD solution was interpolated from the interfaces, Σ 1,2 , to the mesh nodes of the full mesh. The rate of convergence can be estimated from the numerical results as

J k = log e k /e k-1 log ε k /ε k-1 where e k = u ε k -u ε k ,δ H 1 (Bε k ) , ε k = 2 -k , k = 2, ..., 6.
While not constant, for C = 2/ ln(2), J k is in the range 3 J k 6. The error drops at least with cubic convergence (in the investigated cases).

For C = 3/ ln(2) the convergence rate is greatly improved, and even more so using C = 4/ ln(2), namely we obtain J ≈ 8 and J ≈ 11, respectively, discarding the points where the error stagnates. The stagnation of both cases for ε < 2 -4 or 2 -3 is due to the precision of the numerical method being reached. Rounding errors gain importance for very small values of ε.

Transient Navier-Stokes test case

The asymptotic behavior of the error of the MAPDD method with respect to the full model is shown for different Reynolds numbers in Fig. 4, for C = 2/ ln(2). The error is evaluated in the norm [START_REF] Pileckas | Three-dimensional solenoidal vectors[END_REF]. For the lowest investigated Reynolds number Re = 1, the rate of convergence J was computed (omitting the two largest values of ε). The line ε J is included in the figure for comparison. With increasing Reynolds numbers the rate of convergence decreases. Exponential increase of the error was observed for Re = 100. Using C = 3/ ln(2) (see Fig. 5), the rate of convergence obtained for Reynolds numbers Re > 1 is improved. In particular, for Re = 100 the error now decreases with ε, at least for small values of ε.

The errors of the case Re = 100 obtained for C = K/ ln(2), K = 2, 3, 4, are shown in Fig. 6. Indeed, for higher K, the errors are lower and convergence is improved for ε 2 -4 .

While the error estimate assumes a low Reynolds number, the MAPDD method can still be applied to these cases. Figures 7 and8 show velocity streamlines and the pressure field obtained with the full reference model and the MAPDD method applied to the case ε = 1/4 and for a Reynolds number of Re = 2500, as an example. The boundary mesh size was set to h = ε/20, furthermore C = 2/ ln(2). The results match very well visually. The MAPDD model is able to recover the recirculation zones in both junctions accurately (Fig. 7(a) and (b)). For a more detailed

2 -6 2 -5 2 -4 2 -3 2 -2 2 -1 10 -11
10 -9 10 -7

10 -5 comparison, the axial velocity profiles at the interfaces for the MAPDD solution and for the full solution in the corresponding location are shown in Figs. 9. At the left interface, the velocity interface conditions produce a pressure overshoot near the upper corner, since the Womersley hypothesis is in disagreement with the high Reynolds number flow conditions. This can be seen more clearly in Fig. 10a, where the pressure profile at the interface is shown for both the MAPDD and the full solution. However, analyzing the pressure distribution along the cross-section the tube in a slightly more upstream position (shifted upstream by 2ε), the MAPDD recovers the behavior observed for the full solution with an error of < 8% (Fig. 11). The pressure on the right interface does not suffer any nonphysical oscillations, as can be seen in Fig. 10b, and the discrepancy between both models is within 2%.

10 -3 ε u ε -u ε,δ H 1 (Bε) C = 2/ ln 2 C = 3/ ln 2 C = 4/ ln 2 ∝ ε J , J = 4 ∝ ε J , J = 8 ∝ ε J , J = 11

Conclusion

The MAPDD was shown to be an efficient and accurate method for the steady Stokes problem and for the low Reynolds number Navier-Stokes problem. In these cases, the error of the MAPDD method was in agreement with theoretical error estimates, ( 17) and ( 24), respectively. For slightly larger Reynolds numbers, the convergence can be improved by modifying the computational domain and adjusting the constant in Eq. ( 25). Although the theory is only valid for small Reynolds numbers, the method yields very good results also for high Reynolds numbers. For the (arbitrary) example of Re = 2500, ε = 1/4, the MAPDD velocity and pressure solutions were in good agreement with the full solution, except for pressure oscillations that occur near the upstream interface.

2 -6 2 -5 2 -4 2 -3 2 -2 2 -1 10 -3

Appendix 1. Proofs of the main theorems

Consider the steady state Stokes equations [START_REF] Blanc | Asymptotic analysis and partial asymptotic decomposition of the domain for Stokes equation in tube structure[END_REF]. Let us give a weak formulation form equivalent to Definitions 1.1' and 1.2' introducing a new unknown function v ε = u ε -g ε , which is divergence free and vanishing at the whole boundary. Here g ε is a divergence free extension of the boundary values (see Remark 1).

Definition 1.3'.

By a weak solution we understand the vector-field

u ε ∈ H 1 div0(∂Bε\γε) (B ε ) such that the difference v ε = u ε -g ε belongs to H 1 div0 (B ε
) and satisfies the following integral identity: for all v ∈ H 1 div0 (B ε ),

ν Bε ∇v ε (x) : ∇v(x)dx = -ν Bε ∇g ε (x) : ∇v(x)dx . ( 26
)
It is well known that there exists a unique solution to this problem (see [START_REF] Ladyzhenskaya | The Mathematical Theory of Viscous Incompressible Fluid[END_REF]). Further we will need as well a modification of this problem containing a right hand side

f ε = f 0ε -n i=1 ∂f iε ∂x i where f iε ∈ L 2 (B ε ), i = 0, 1, ..., n:              -ν∆v ε + ∇p ε = f ε , x ∈ B ε , divv ε = 0, x ∈ B ε , v ε = 0, x ∈ ∂(B ε ) , ( 27
)
with variational formulation: to find a vector-valued function This problem as well admits a unique solution and using the well-known Poincaré-Friedrichs inequality with a constant proportional to ε (see for example [START_REF] Panasenko | Multi-scale Modeling for Structures and Composites[END_REF]) we get an a priori estimate:

v ε ∈ H 1 div0 (B ε ) such that for any test function v ∈ H 1 div0 (B ε ) it satisfies the following integral identity: for all v ∈ H 1 div0 (B ε ), ν Bε ∇v ε (x) : ∇v(x)dx = Bε f 0ε • v(x) + n i=1 f iε • ∂v ∂x i (x)dx . ( 28
) 2 -6 2 -5 2 -4 2 -3 2 -2 2 -1 10 -3
v ε H 1 (Bε) ≤ C n i=0 f iε L 2 (Bε) , ( 29
)
where C does not depend on f ε .

Divergence free extension of boundary value function having the steady Poiseuille shape in the cylinders

Let us prove that there exists a special divergence free extension g ε such that within some interior part of the cylinders

B (e)
ε it is a Poiseuille flow [START_REF] Kapitanskii | Certain problems of vector analysis[END_REF] . Let us remind the definition of the steady Poiseuille flow in a cylinder B (e) ε . Lemma A1.Function g ε can be extended to B ε so that its extension

g ε ∈ H 1 (B ε ), in each subdomain ωj ε = ω j ε (B (e) ε ∩ {x (e)
1 < εd}) (here e is the edge containing O j , j = N 1 + 1, ..., N ) it is equal to

g ε (x) = g j ( x -O j ε ),
and for any cylinder

B (e) ε ∩ {3εd < x (e) 1 < |e| -3εd} is equal in local coordinates to a Poiseuille flow V P (x).
Proof:

Let us consider a problem for the pressure p on the graph B (in this proof we omit for simplicity the index ε) (see [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF]) with the continuity condition for the pressure on the graph and with given

2 -6 2 -5 2 -4 2 -3 2 -2 2 -1 10 -1 10 0 ε Velocity error C = -2/ ln(2) C = -3/ ln(2) C = -4/ ln(2) ∝ ε J , J = 0.54
κ j ∂ 2 p ∂x (e j ) 2 1 = 0, x ∈ e j , j = 1, ..., M, j:e j ∈O i κ j ∂p ∂x (e j ) 1 = 0, i = 1, ..., N 1 , κ j ∂p ∂x (e j ) 1 = γ j ε g ε • nds, i = N 1 + 1, ..., N,
p is continuous f unction on the graph B.

Here κ j = ε n-1 σ j v P (y )dy , and the local axes have the origin O i . This problem admits a unique (up to an additive constant) solution, linear function on every edge, see [START_REF] Panasenko | Flows in a tube structure: equation on the graph[END_REF]. The slope of this solution at each edge defines the Poiseille flow in the corresponding cylinder: V (e j ) P , such that

κ j ∂p ∂x (e j ) 1 = σ jε V (e j ) P
• nds,

σ jε = {x ∈ B ε j |x 1 = 0}.
Then for every domain ωj ε we construct a divergence free extension equal to the determined Poiseuille flows on the parts of the boundary coinciding with the sections of the cylinders. The possibility of this construction is insured by the flux balance in every node of the graph (see [START_REF] Ladyzhenskaya | On some problems of vector analysis and generalized formulations of boundary value problems for the Navier-Stokes equations[END_REF], [START_REF] Pileckas | Three-dimensional solenoidal vectors[END_REF], [START_REF] Pileckas | On spaces of solenoidal vectors[END_REF], [START_REF] Bogovskii | Solutions of some problems of vector analysis related to operators div and grad[END_REF]). 2

Lemma A2. The extension may be constructed in such a way that it belongs to H 2 (B ε ).

Proof:

Indeed, for every edge e i multiply constructed Poiseuille flow V

(e i )
P by a cut-off function ζ(

x

(e i ) 1 3dε )ζ( |e i | -x (e i ) 1 3dε ), where ζ(τ ) is a smooth cut-off function independent of ε with ζ(τ ) = 0 for τ ≤ 1/3 and ζ(τ ) = 1 for τ ≥ 2/3, 0 ≤ ζ(τ ) ≤ 1. Denote ψ ε = div (ζ( x (e i ) 1 3dε )ζ( |e i | -x (e i ) 1 3dε )V (e i ) P (x)). Evidently, ψ ε ∈ H 1 0 (B ε )
and Bε ψ ε dx = 0. Then for every domain ωj ε we can construct a functionw ε such that div w ε = -ψ ε and w ε ∈ H 2 0 (ω j ε ), j = 1, ..., N 1 (see [START_REF] Bogovskii | Solutions of some problems of vector analysis related to operators div and grad[END_REF], [START_REF] Kapitanskii | Certain problems of vector analysis[END_REF]) and w ε ∈ H 2 0 (ω j ε ), j = N 1 + 1, ..., N , moreover, w ε and its gradient vanish on the part of ∂ ωj ε belonging to B ε , and w ε = g ε on γ j ε . We take gε = w ε + ζ( x

(e i ) 1 3dε )ζ( |e i | -x (e i ) 1 3dε
)V

(e i ) P (x).2

Estimate for the difference between the exact solution and the MAPDD solution: asymptotic analysis of the Stokes equations

Theorem 1. Given natural number J there exists a constant C (independent of ε and J) such that if δ = CJε| ln ε|, then

u ε -u ε,δ H 1 (Bε) = O(ε J ) . (30)
Proof.

1. Consider an asymptotic expansion of the solution (see [START_REF] Panasenko | Asymptotic expansion of the solution of Navier-Stokes equation in a tube structure[END_REF], [START_REF] Blanc | Asymptotic analysis and partial asymptotic decomposition of the domain for Stokes equation in tube structure[END_REF]). For the velocity u ε it has the following shape: a Poiseuille flow V P (x) within the cylinders B (e) ε plus some functions depending on the variable

x -O j ε exponentially tending to zero in the cylinders as the variable x -O j ε tends to infinity. One can write this expansion as a uniform approximation of order J in a form

v (J) ε (x) = M i=1 ζ( x (e i ) 1 3dε )ζ( |e i | -x (e i ) 1 3dε )V [i,J] P (x (e) /ε) + N l=1 (1 -ζ( |x -O l | |e| min ))V [BLO l ,J] ( x -O l ε ). (31)
Here V

[i,J] P (y (e) ) = C [i,J] (k e1 v P (y (e) ), ..., k en v P (y (e) )) T (as above, e has the cosines directors k e1 , ..., k en and the local variables x (e) are related to the global ones by the equation x (e) = x (e) (x)), C [i,J] are constants such that for any node O l the flux conservation law is satisfied:

i:O l ∈e i C [i,J] σ (e i ) v P (y (e) )dy (e) = 0 (32)
(the local coordinate system x (e i ) has its origin in O l ) and for vertices O l , the end points of only one edge e i ,

C [i,J] σ (e i ) v P (y (e) )dy (e) + γ ε l g ε • nds ε 1-n = 0 ( 33
)
(n is an outer normal vector). Every V [i,J] P (y (e) ) is defined only within the cylinder B (e) ε associated to the edge e. Here |e| min is the minimal length of the edges. The boundary layer term, vector valued function V [BLO l ,J] (y), exponentially vanishes as |y| tends to infinity: in the sense that the product V [BLO l ,J] (y)e β|y| belongs to the space H 1 (Ω l ) with some positive β independent of ε,

Ω l = {y ∈ R n |ε(y + O l ) ∈ Bε,δ l }, and Bε,δ l is an extension of B ε,δ l behind the cross-sections S ij as semi-infinite cylinders: Bε,δ l = B ε,δ l ∪ e:O l ∈e {x (e)
1 > εd; x (e) /ε ∈ σ (e) }. So, Ω l is an unbounded domain obtained from the bounded domain {ε(y+O l ) ∈ B ε,δ l } with truncated cylinders by extention of them behind the truncations, so that they become the cylindrical outlets to infinity. Functions V [BLO l ,J] satisfy the integral identity: for all divergence free vector valued test functions Φ from

Ω l ν∇V [BLO l ,J] (y) : ∇Φ(y)dy + i:O l ∈e i Ω l ν∇ V [i,J] P (y (e) )ζ( y (e i ) 1
H 1 0 (Ω l ) and div V [BLO l ,J] (y) + i:O l ∈e i V [i,J] P (y (e) )ζ( y (e i ) 1 3d ) = 0.
V [BLO l ,J] satisfy the homogeneous Dirichlet boundary conditions if O l is an end point for at least two edges:

V [BLO l ,J] (y) = 0, y ∈ ∂Ω l ; (35)
or the non-homogeneous Dirichlet boundary conditions

V [BLO l ,J] (y) = g l (y), y ∈ γ l ; V [BLO l ,J] (y) = 0, y ∈ ∂Ω l \γ l ; (36) in the case if O l is a vertex, γ l = ∂Ω l ∩ ∂ω l .
For the asymptotic expansion the following estimate (see [START_REF] Panasenko | Multi-scale Modeling for Structures and Composites[END_REF], [START_REF] Blanc | Asymptotic analysis and partial asymptotic decomposition of the domain for Stokes equation in tube structure[END_REF])

u ε -v (J) ε H 1 (Bε = O(ε J ) (37)
holds.

2. Let us multiply all boundary layers by a cut-off function passing from value one within the distance less than δ/3 from the nodes to the value zero if the distance from the nodes is greater than 2δ/3, i.e. we replace v (J) (x) by the new asymptotic approximation

u a ε (x) = M i=1 ζ( x (e i ) 1 3dε )ζ( |e i | -x (e i ) 1 3dε )V [i,J] P (x (e) /ε) + N l=1 (1 -ζ( |x -O l | δ ))V [BLO l ,J] ( x -O l ε ). ( 38
)
This new approximation consists only from the Poiseille flow within the cylinders B Let us choose δ equal C J ε| ln ε||e| min and choose the constant C J such that the residual in the right-hand side of the equations has the order O(ε J ). To this end notice that the boundary layer functions decay exponentially, i.e. V [BLO l ,J] (y)e β|y| belongs to the space H 1 (Ω l ) with some positive β independent of ε. Let us find δ such that

F l, δ ε = O(ε J+2 ) ( 39
)
where

F l,R = V [BLO l ,J] H 1 (Ω l,R ) ,
and Ω l,R = Ω l ∩ {|y| > R}. Indeed, the inclusion V [BLO l ,J] (y)e β|y| ∈ H 1 (Ω l ) implies

e β δ ε F l, δ ε ≤ e β|y| V [BLO l ,J] (y) H 1 (Ω l, δ ε ) ≤ e β|y| V [BLO l ,J] (y) H 1 (Ω l )
and the last norm is bounded by a constant, denote it C l . So, we can write

e β δ ε F l, δ ε ≤ C l , i.e. F l, δ ε ≤ C l e -β δ ε .
Let us take δ = 1 β (J + 2)ε| ln ε|, then we get the estimate (39), and so, making the change of variables x-O l = εy, we get that the difference v (J) -u a ε has support belonging to the cylinders B

) = O(ε J ). Thus, v (J) -u a ε H 1 (Bε) = O(ε J ).
3. Unfortunately, u a ε may be not divergence free within the parts B

1 -ζ( |x -O l | δ ))V [BLO l ,J] ( x -O l ε
) have the divergence equal to

h l (x) = -div ζ( |x -O l | δ )V [BLO l ,J] ( x -O l ε ) = -δ -1 ζ ( |x -O l | δ )∇|x -O l | • V [BLO l ,J] ( x -O l ε ).
The vector field V [BLO l ,J] belongs to H 1 0 (Ω l ) and, therefore, the flux of it in every outlet to infinity is equal to zero. Let us show that

B (e),δ/3,± ε h l (x)dx = 0. Indeed, B (e),δ/3,± ε div ζ( |x -O l | δ )V [BLO l ,J] ( x -O l ε ) dx = ∂B (e),δ/3,± ε ζ( |x -O l | δ )V [BLO l ,J] ( x -O l ε ) • nds = = σ (e) ε (x (e) 1 =2δ/3) V [BLO l ,J] ( x -O l ε ) • ndS = 0,
where σ ε .

( |x -O l | δ ) = 0 for |x -O l | δ ≤ 1/3, ζ( |x -O l | δ ) = 1 for |x -O l | δ ≥ 2
Thus, we can apply the results from [START_REF] Panasenko | Divergence equation in thin-tube structure[END_REF] (see Lemma 3.1 and the change of variables from Lemma 3.6 in [START_REF] Panasenko | Divergence equation in thin-tube structure[END_REF]) and prove that there exists w ∈ H 1 0 (B (e),δ/3,± ε

) such that divw = -h l (x) and

h l L 2 (B (e),δ/3,± ε ) ≤ c δ V [BLO l ,J] L 2 (B (e),δ/3,± ε ) = O( 1 δ ε J+2+s/2 )
and so,

w H 1 (B (e),δ/3,± ε ) = O(ε J ).
Extending w by zero outside the domains B (e),δ/3,± ε

we get an approximation u (J) ε = u a ε + w which belongs to the space H 1,δ div0(∂Bε\γε) (B ε ). 4. Evidently the difference u (J) ε -u ε satisfies the homogeneous Dirichlet boundary condition on ∂B ε and the Stokes equations in B ε with a residual of order O(ε J ). There holds the following integral identity

ν Bε ∇(u (J) ε (x) -u ε (x)) : ∇v(x)dx = -ν B ε ∇r ε (x) : ∇v(x)dx ∀v ∈ H 1 div0 (B ε ), (40)
where

r ε (x) = ∇(w(x) + (u a ε (x) -v (J) (x)) + (v (J) (x) -u ε (x))
). Due to the previous estimates the norms w H 1 (Bε) , u a ε -v (J) H 1 (Bε) and v (J) -u ε ) H 1 (Bε) are of order O(ε J ) and we get that

r ε L 2 (Bε) = O(ε J ).
So, applying the a priori estimate (29), we get the following inequality

u ε -u (J) ε H 1 (Bε) = O(ε J ) . ( 41
)
5. Consider now the projection of problem (6) on the subspace H 1,δ div0(∂Bε\γε) (B ε ). By the Lax-Milgram theorem there exists a unique solution u ε,δ to this projection and the difference u (J) ε -u ε,δ belongs to the space H 1,δ div0(∂Bε\γε) (B ε ) and satisfies, as before (see (40)), variational formulation with a residual of order O(ε J ): for every vector field

v ∈ H 1,δ div0 (B ε ), ν Bε ∇(u (J) ε (x) -u ε,δ (x)) : ∇v(x)dx = -ν B ε ∇r ε (x) : ∇v(x)dx. (42)
Now applying an a priori estimate (29) we get

u (J) ε -u ε,δ H 1 (Bε) = O(ε J ) . ( 43
)
Estimates (37), (43) imply (30). 2 Remark A1.Notice that in [START_REF] Panasenko | Asymptotic expansion of the solution of Navier-Stokes equation in a tube structure[END_REF], [START_REF] Blanc | Asymptotic analysis and partial asymptotic decomposition of the domain for Stokes equation in tube structure[END_REF], [START_REF] Panasenko | Multi-scale Modeling for Structures and Composites[END_REF]

(ε J ) in the L 2 (B ε )-norm: -ν∆u (J) ε (x) + ∇p a ε (x) = r ε (x), x ∈ B ε , ( 44
)
where

r ε L 2 (Bε) = O(ε J ).
The boundary conditions are satisfied exactly and divu (J) ε = 0.

Navier-Stokes equations

Consider the Navier-Stokes equations [START_REF] Alnaes | The FEniCS Project Version 1.5[END_REF]. Let us remind a weak formulation from [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF] which is equivalent to Definitions 1.1 and 1.2. It introduces a new unknown function v ε = u ε -g ε , which is divergence free and vanishing at the whole boundary. Definition 1.3. By a weak solution we understand the vector-field

u ε = v ε + g ε , where divv ε = 0, v ε (x, 0) = 0, v ε ∈ L ∞ (0, T ; H 1 0 (B ε )), v t ∈ L 2 (0, T ; L 2 (B ε ))
, and for every vector-field φ ∈ H 1 0 (B ε ) and for all t ∈ (0, T ), v ε satisfies the integral identity

Bε v εt • φ + ν∇v ε • ∇φ -(v ε + g ε ) • ∇ φ • v ε -v ε • ∇ φ • g ε dx = -ν Bε ∇g ε • ∇φdx - Bε g ε • ∇ g ε + g εt • φdx. ( 45
)
The proof of the existence of a solution for sufficiently small values of ε repeats literally the proof from [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF].

The new version of the method of asymptotic partial domain decomposition (MAPDD) replaces the problem (1) by [START_REF] Ladyzhenskaya | On some problems of vector analysis and generalized formulations of boundary value problems for the Navier-Stokes equations[END_REF], where the space H 1,δ div0(∂Bε\γε) (B ε ) is replaced by the newly defined space of divergence free vector-functions having the Womersley form within cylinders B dec,ε ij . Let us justify the weak MAPDD formulation with pressure [START_REF] Panasenko | Flows in a tube structure: equation on the graph[END_REF]. Lemma A3. Let u ε,δ be a solution to problem [START_REF] Panasenko | Partial asymptotic decomposition of domain: Navier-Stokes equation in tube structure[END_REF]. For all test functions φ ∈ H 1,δ 0 (B ε ),

N i=1 j:O i O j ∈{e 1 ,...,e M } ∂B ε,δ i ∩S ij ν ∂u ε,δ ∂n -p ε,δ n • φds + + M l=1 d l σ (e l ) ε u ε,δ,t • φ + ν∇ x (e l ), u ε,δ : ∇ x (e l ),
φdx (e l ), = 0.

Proof.

On the other hand the first component of φ is an arbitrary function with vanishing mean. So, in every B dec,ε ij we get an equation for u ε,δ,1 , the first component of u ε,δ :

(|e ij | -2δ) u ε,δ,1,t -ν∆u ε,δ,1 + (σ n (u ε,δ , p ε,δ,i )| S ij + σ n (u ε,δ , p ε,δ,j )| S ji ) • e ij = D ij (t), (46)
where D ij (t) are "constants" depending on time only. 2. Let us take now an arbitrary test function from the space H 1,δ div0 (B ε ) and again take into account vanishing of terms B ε,δ i p ε,δ,i divφdx, we get:

i,j:i<j,O i O j ∈{e 1 ,...,e M } B dec,ε ij D ij (t)φ 1 dx = 0, (47)
i.e.

i,j:i<j,O i O j ∈{e 1 ,...,e M } D ij (t) B dec,ε ij φ 1 dx = 0. Denote F ij = S ij φ 1 ds = (|e ij | -2δ) -1 B dec,ε ij φ 1 dx.
Varying φ we get that for any arbitrary set of fluxes F ij satisfying equation

j:e ij ⊂B F ij = 0 (48)
for all i = 1, ..., N, the following relation holds:

j:e ij ⊂B (|e ij | -2δ)D ij (t)F ij = 0. ( 49
)
3. Consider now an arbitrary function from the space H 1,δ 0 (B ε ). Remind that such functions satisfy condition [START_REF] Panasenko | Multi-scale Modeling for Structures and Composites[END_REF] but no more divergence free in B ε,δ i . Consider the expression in the assertion of the Lemma:

N i=1 j:O i O j ∈{e 1 ,...,e M } ∂B ε,δ i ∩S ij ν ∂u ε,δ ∂n -p ε,δ n • φds+ + M l=1 d l σ (e l ) ε u ε,δ,t • φ + ν∇ x (e l ), u ε,δ : ∇ x (e l ), φdx (e l ), .
Using now relations (46), we get that this expression is equal to

i,j:i<j,O i O j ∈{e 1 ,...,e M } B dec,ε ij D ij (t)φ 1 dx = i,j:i<j,e ij ⊂B (|e ij | -2δ)D ij (t)F ij .
Note that test functions satisfy relations [START_REF] Panasenko | Multi-scale Modeling for Structures and Composites[END_REF] which implies (48), and so, relations (49) hold. Lemma A3 is proved.

Remark A2. Note that in the case N = M + 1 condition ( 19) can be removed from the definition of the reduced space, and we can take D ij = 0 without condition (48), only controling constants θ i . Indeed, in (46) every stress σ n (u ε,δ , p ε,δ,i )| S ij is defined up to an addidive constant θ i (t), so relations (46) define uniquely some constants F ij (t) equal to θ i (t)-θ j (t)-D ij (t). Consider the following system of equations for θ i :

θ i (t) -θ j (t) = F ij (t), i < j, e ij ⊂ B. ( 50
)
Setting θ N = 0, we get the system of N -1 equations with non-degenerate matrix. Indeed, the homogeneous system is θ i = θ j , i < j, e ij ⊂ B. It means that due to the conectedness of the graph B the unique solution to the homogeneous system is: all θ i = 0. So, system (50) admits a unique solution such that θ N = 0. Consequently, constants θ i can be chosen in such a way that D ij = 0. This choise of θ i is unique up to one constant θ N . So, integral identity [START_REF] Panasenko | Flows in a tube structure: equation on the graph[END_REF] holds true for a wider subspace of test functions with removed condition [START_REF] Panasenko | Multi-scale Modeling for Structures and Composites[END_REF].

Estimate for the difference between the exact solution and the MAPDD solution for the non-stationary Navier-Stokes equations

The result of the previous section can be generalized for the non-stationary problem for the Navier-Stokes equations (1) using the approach of [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF] and [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure[END_REF]. Assume that g ε introduced in the second section has a divergence free extension gε defined in B ε with g ∈ C [

where the constant c is independent of ε.

As before,

∂Bε g ε • nds = 0. ( 52
)
The proof of the existence of the unique solution to (1) for sufficiently small values of ε repeats literally the proof from [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF].

A complete asymptotic expansion of order J to this problem is constructed in [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF], [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure[END_REF] and it has the form

v (J) ε (x, t) = M i=1 ζ( x (e i ) 1 3dε )ζ( |e i | -x (e i ) 1 3dε
)(V [i,J,N SI] P (x (e) /ε, t) + ε 2 V [i,J,N SII] P (x (e) /ε, t/ε 2 ))+

+ N l=1 (1 -ζ( |x -O l | |e| min ))(V [BLO l ,J,N SI] ( x -O l ε , t) + ε 2 V [BLO l ,J,N SII] ( x -O l ε , t/ε 2 )), (53)
where the functions V [i,J,N SI] P and V

[i,J,N SII] P are C 2 -smooth functions such that in local variables only the longitudinal component of the velocity may be different from zero and its longitudinal derivative is equal to zero, while V [BLO l ,J,N SI] and V [BLO l ,J,N SI] are the boundary layer correctors belonging to the spaces L 2 ((0, T ); H 1 (Ω l )) and L 2 ((0, ∞); H 1 (Ω l )) respectively, and such that for some positive β, e β|y| V [BLO l ,J,N SI] and e β|y| V [BLO l ,J,N SII] also belong to these spaces. The corresponding norms of these four terms are bounded by constants independent of ε. The vector function v (J) ε satisfies the estimate sup t∈(0,T ) v (J) ε -u ε L 2 (Bε) + ∇v (J) ε -∇u ε L 2 ((0,T );L 2 (Bε)) = O(ε J-2 )

For more details see [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF], [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure[END_REF]. Remark A3. An asymptotic expansion for the pressure was as well constructed in [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF], [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure[END_REF] and has a similar structure:

p (J) ε (x, t) = M i=1 ζ(
x 1 , while P [BLO l ,J,N SI] and P [BLO l ,J,N SII] are the boundary layer terms belonging to the spaces L 2 ((0, T ); L 2 (Ω l )) and L 2 ((0, ∞); L 2 (Ω l )) respectively and such that for some positive β, e β|y| P [BLO l ,J,N SI] and e β|y| P [BLO l ,J,N SII] also belong to these spaces. (Standard theorems on the asymptotic behavior of the pressure in domains with outlets at infinity establish that it tends to some constants, however the construction of asymptotic expansion in [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF] subtracts these constants so that the pressure terms belong to the space L 2 ). The couple (v

(
(J) ε , p (J)
ε ) satisfies the Navier-Stokes equations in classical sense with a residual of order O(ε J-2 ) in the H 1 ((0, T ); L 2 (B ε ))-norm. The boundary and the initial conditions are satisfied exactly.

Consider the following setting: to find v ε,δ ∈ L 2 (0, T ; H 1,δ div0(∂Bε\γε) (B ε )) such that u ε,δ = v ε,δ + gε , v ε,δ (x, 0) = 0, v ε,δ ∈ L 2 (0, T ; H 1,δ div0 (B ε )), v ε,δ,t ∈ L 2 (0, T ; L 2 (B ε )), and v ε,δ satisfies the integral identity

Bε v ε,δ,t • Φ + ν∇v ε,δ • ∇Φ -(u ε + gε ) • ∇ Φ • v ε,δ -v ε,δ • ∇ Φ • gε dx = -ν Bε ∇g ε • ∇Φ dx - Bε gε • ∇ gε + gεt • Φ dx (55)
for every divergence free vector-field Φ ∈ H 1,δ div0 (B ε ). Here an extension gε is such that it belongs to L 2 (0, T ; H 1,δ div0(∂Bε\γε) (B ε )). Existence and uniqueness of a solution to an analogous problem set in the usual Sobolev spaces is proved in [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF] by Galerkin method. For the problem (55) the proof is just the same. Note only that an orthogonal base exists due to the separability of the space H 1,δ div0(∂Bε\γε) (B ε ) (as a subspace of a separable space).

Theorem 2.Let g j ∈ C [ J+4 2 ]+1 ([0, T ]; W 3/2,2 (∂ω j )). Given natural number J there exists a constant C (independent of ε and J) such that if δ = CJε| ln ε|, then sup t∈(0,T ) u ε,δ -u ε L 2 (Bε) + ∇(u ε,δ -u ε ) L 2 ((0,T );L 2 (Bε)) = O(ε J ) .

(56)

2 LFigure 1 :

 21 Figure 1: Illustration of the computational domain for N = 2 and M = 1.

  divergence free vector-valued functions vanishing at the whole boundary. Definition 1.1.

3. 1

 1 The reduced domain and classical version of MAPDD Let us remind first the definition of the steady Poiseuille flow in a cylinder B (e) ε . If the local variables x (e) for the edge e coinside with the global ones x then the Poiseuille flow is defined as V (e) P (x) = const (v P (x /ε), 0, ..., 0) T , where v P (y) is a solution to the Dirichlet's problem for the Poisson equation on σ (e) :

)C 2 =

 2 is a user parameter. Pairs of full and reduced domains are created for a sequence of values ε = 2 -k , k = 1, ..., 6. In the particular examples of the investigated geometry and our selection of ε, 1/ ln(2) < C < 6/ ln(2) is necessary for B ε,δ 1;2 = ∅ and for B dec,ε 1,∅, respectively. In what follows, we choose the values C = K/ ln(2), K = 2, 3 and 4. The factor 1/ ln(

Figure 2 :

 2 Figure 2: Pressure fields and velocity magnitude and vectors at the outflow boundaries obtained for the stationary Stokes problem using ε = 0.5 with the full model (top row) and with the MAPDD model (bottom row).

Figure 3 :

 3 Figure 3: Stationary Stokes test case: convergence of the error with respect to ε for different values of C (see legend).

Figure 4 :

 4 Figure 4: (Eq. (24)) of the Navier-Stokes MAPDD model w.r.t. to the full solution for different Reynolds numbers, with C = 2/ ln(2); the line ε J=0.45 for reference.

4 Figure 5 :

 45 Figure 5: Errors (Eq. (24)) of the Navier-Stokes MAPDD model w.r.t. to the full solution for different Reynolds numbers, with C = 3/ ln(2); the line ε J=0.4 for reference.

Figure 6 :

 6 Figure 6: Comparison of the Navier-Stokes error with different values of C for Re = 100.

  (a) Velocity -Full order solution (b) Velocity -MAPDD solution

Figure 7 :

 7 Figure 7: Velocity stream lines of the transient Navier-Stokes test case at peak time t = 0.2 s, for Re = 2500, ε = 0.25. Full model (a) versus MAPDD model (b).

Figure 8 :

 8 Figure 8: Pressure fields of the transient Navier-Stokes test case at peak time t = 0.2 s, for Re = 2500, ε = 0.25. Full model (a) versus MAPDD model (b).

Figure 9 :

 9 Figure 9: Axial velocity component u 0 at the interfaces for the MAPDD and the full solutions computed for Re = 2500, ε = 1/4.

Figure 10 :

 10 Figure 10: Pressure along the interfaces for the MAPDD and the full solutions computed for Re = 2500, ε = 1/4.

Figure 11 :

 11 Figure 11: Pressure along the tube cross-section, at 2ε upstream of Σ 1 , for the MAPDD and the full solutions computed for Re = 2500, ε = 1/4.

1 >

 1 |e| -2δ/3} of the cylinders B (e),δ/3 ε because the products

(

  /3, and we have used that ζ

  /3, and that V [BLO l ,J] is equal to zero on the lateral boundary of B (e)

J+4 2 ]-1 2 ;∇g L 2 (-3 2 ,∆g L 2 ( 5 2

 222225 +1 ([0, T ]; H 2 (B ε )) satisfying for all t ∈ [0, T ] the following asymptotic estimates g L 2 (Bε) + gt L 2 (Bε) + gtt L 2 (Bε) ≤ cε nBε) + ∇g t L 2 (Bε) ≤ cε nBε) ≤ cε n-, n = 2, 3,

  For the linear pressure the second order differential equation on the graph with the Kirchhoff type junction conditions in the nodes hold. The couple (v ) satisfies equations[START_REF] Blanc | Asymptotic analysis and partial asymptotic decomposition of the domain for Stokes equation in tube structure[END_REF] in classical sense with a residual of order O(ε J ) in the L 2 (B ε )-norm. Moreover, the couple (u

	longitudinal variable for each cylinder B	(e) ε multiplied by a cut-off function plus the boundary layer
	terms exponentially decaying as tends to infinity. (J) |x -O l | ε ε , p (J)
		(J) ε , p a ε ), where p a ε is obtained from p (J) ε	replacing the cut-off
	factor of boundary layers by 1 -ζ(	|x -O l | δ	), as well satisfies equations (6) in classical sense with
	a residual of order O
				(J) ε	of the pressure p ε
	was as well constructed and it has similar to (31) structure: a linear pressure depending on the

an asymptotic expansion p ε

The full dimensional fluid flow problem in a tube structureIn this section we will introduce the full dimensional fluid flow problem in a tube structure. Further its solution will be approximated using partial dimension reduction.

1. Consider in [START_REF] Panasenko | Partial asymptotic decomposition of domain: Navier-Stokes equation in tube structure[END_REF] the divergence free test functions φ vanishing in all cylinders B dec,ε ij except for one of them and two adjacent domains B ε,δ i and B ε,δ j . We get

Integrating by parts and taking into account that in cylinders B dec,ε ij functions u ε,δ and φ are independent of the longitudinal variable, we get:

respectively. So,

Integrate the terms with pressure by parts:

Here

where e ij is the vector director of Ōi O j .

Proof:

The idea of the proof is similar to that of Theorem 1. Replace v

x

where δ = C J ε| ln ε||e| min and C J is chosen in such a way that

and

Namely, C J = 9(J + 2) β|e| min , see [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF], p.158. (In fact, the order 3J + 6 is too much for the estimate of Theorem 2, J + 2 is enough 1 ). Then we use the same arguments as in the proof of Theorem 6.2 of [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF]: we check that this new approximation v (J+2,a) ε with the modified in the same way approximation p (J+2,a) ε satisfy the Navier-Stokes equations in the classical sense with a residual of order O(ε J ) in the norm H 1 ((0, T ); L 2 (B ε )), and that the velocity approximation satisfies the boundary and initial conditions exactly. In the same way the pair (v

) satisfies the problem (55) with the same error. Applying results from the paper [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF] (see the inequality (5.7) in the proof of Theorem 5.1 in [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF]), we get the estimate sup t∈(0,T )

Applying the estimate sup t∈(0,T )

and the triangle inequality we derive (56). 2

Acknowledgments

The work was funded by the European Social Fund according to the activity "Improvement of researchers qualification by implementing world-class R and D projects" of Measure No. 09.3.3-LMT-K-712. The first and second authors were partially supported by PFBasal-001 and AFBasal170001 projects.

1 Note that in Remark 5.1 of the paper [START_REF] Panasenko | Asymptotic analysis of the non-steady Navier-Stokes equations in a tube structure. I. The case without boundary layer-in-time[END_REF]


In order to design an Intelligent Instructional System (IIS), the designer has to know w hat instructional theories and m odels m ay provide appropriate principles and strategies, w hat functional com ponents are necessary for reaching instructional goals and obj ectives, w hat pedagogical actions are appropriate for each situation, w hat architecture and strategy are appropriate for supporting these actions, how to control the system behavior to achieve a coherent learning support process, and how to organize dom ain know ledge. H ow ever, these fundam ental characteristics of an IIS are often im pl i cit, vague or ill-form ed. D espi te m uch theoretical research and im pl em entation of IISs, there is little to link the tw o, and relations betw een research and im pl em entation are not strong. W hat i s needed is a w ell-form ed system of concepts w hich sum m arizes w hat w e have learned to date and characterizes areas of agreem ent, as w el l as areas w here disagreem ents indicate the need for further elaboration. A solution to this problem could be the ontological engineering of instructi on.

A n ontology consists of a task ontology w hich characterizes the com putational architecture of a know ledge-based system w hich perform s a task, and dom ain ontology w hich characterizes the dom ain know l edge w here the task is perform ed, such as diagnosis, m oni toring, scheduling, or design. Instruction is a task, as is supporting the learning process. Task ontology m i ght provide an effective m ethodology and vocabulary for both analyzing and synthesi zing know ledge-based system s to w hich IISs belong, w ith benefits such as: a com m on vocabulary, m aki ng know ledge explicit, system atization, standardization, and m eta-m odel functi onality. This functionality suggests the possibility of an ontology-aw are authoring functionality w hich could be very intel ligent in the sense that it w ould know w hat m odel w ould help authors. M i zoguchi [1] has proposed the follow ing three levels of ontologies. Level 1 is a structured collection of term s. The m ost fundam ental task in ontology developm ent is articulation of the w orld of interest; that is, elicitation of concepts and identifying a is-a hierarchy am ong them . Level 2 adds form al defini tions to prevent unexpected interpretation of the concepts and necessary relations and constraints also form ally defined as a set of axiom s. D efi nitions are declarative and form al to enable com puters to interpret them . The interpretability of an ontology at this level enables com puters to answ er questions about the m odel s built based on the ontology. In level 3, an ontology becom es executable in the sense that m odels built based on it run using m odul es provided by som e of the abstract codes associated w ith concepts in the ontology. Thus, it can answ er questions about runtim e perform ance of the m odel s.

Since an IIS needs term s/ concepts concerning pedagogical actions to ground the functionality in concrete actions, the justification should be given by theories, and the source of intelli gence of the system s should com e from the know ledge bases containing this know ledge. E asy access to educational theories w ould be valuable to both to hum an and com puter agents. For hum ans, conventional brow sers are enough. For com puters, som ew hat deeper operationality is required. O ntological E ngineering hel ps specify higher level functionality of IISs: it bridges the gap betw een hum an know ledge and know ledge in the know ledge bases. A n O ntology of Instruction could pave the w ay for the building of an ID -aw are A uthoring E nvironm ent for IISs. A n authoring agent coul d explain relevant theories in response to an authors request; it could give the author som e possible justifications for teaching and learning strategies from a theoretical point of view . A n Instructional know ledge server on the W eb could have such a support functionality, and be called an Instructional O ntology-aw are environm ent . In order to reach this goal, a first step is to extract an ontology from existing Instructional theories and from Instructional D esign m odel s.

The first challenge is to have com puters m ediate the sharing of our know ledge, w i th a com m on vocabulary for representing the know ledge, in order to do m eani ngful m edi ation using com m on term s. The level 1 ontology plays a sufficient role for this goal, w hich is to share prim i tive concepts in term s that can describe the know ledge and theories. Instructional know ledge could be described in term s of a shared vocabulary, based on an on-line glossary (http: / / garnet. acns. fsu. edu/ ~w w w 6982/ glossary. htm l ), on descriptions of theories (http: / / w w w . cudenver. edu/ ~m ryder/ itc_data/ theory. htm l ), and on taxonom ies of ID know ledge [START_REF] Seels | Taxonom ic Issues and the D evelopm ent of Theory in Instructional[END_REF]. The second challenge is to extend this sharing from am ong hum ans to am ong com puters. Level 2 introduces definitions of each term and relations richer than in level 1 by using axiom s. A n axi om relates a couple of concepts sem antically, w hich m akes com puters partially understand the rationale of the configuration of the w orld of interest, here learning and instructi on. The operationalization of this know ledge leads to the building of IISs. This requires a level 3 ontology to enable com puters run the code corresponding to the activity-related concepts. K now l edge at this level is m ai nl y concerned w ith task ontology w hich contains concepts of action of the system in perform ing a specifi c task (instruction, learning support). The know ledge server com m unicates w ith hum ans w ho need help in finding know ledge appropriate for their goals. Thus, such authoring environm ents can discuss w ith authors about the appropriateness of strategies adopted w ith the hel p of the know ledge server. Future IISs developed in this w ay w ould behave in a seam less flow of know ledge from designers onto learners.

B ui lding an O ntology of Instruction requires us to identify the concepts that w il l constitute this ontology. Instructional Science (IS) consists of theories, m odel s and m ethodologies for Instruction and for R esearch on Instruction; it builds upon Learning Sciences, C ognitive Sciences and System s Science. Instructional Science is a D esign Science, as defined by Sim on, and it has both descriptive and prescriptive com ponents; the prescriptive part form s w hat is called Instructional D esign. Instructional D esign (ID ) is a system ic and system atic process of applying strategies and techniques derived from behavioral, cognitive, and constructivist theories to the solution of instructional problem s; it represents the system atic application of theory and other organized know ledge to the task of instructional design and developm ent (http: / / w w w . unc. edu/ cit/ guides/ irg-22. htm l). ID is dom ain-independent, generic, theory-based; it contains concepts, rules and principles. The state of the art in ID show s concerns about unification and integration [START_REF] Seels | Instructional D esi gn Fundam entals: a R econsideration[END_REF][START_REF] Uchastel | Prolegom ena to a theory of instructional design[END_REF], as w ell about taxonom ic issues [START_REF] Seels | Taxonom ic Issues and the D evelopm ent of Theory in Instructional[END_REF], tow ard a better integration of taxonom ic concepts betw een learning dom ains -affective, cognitive and psychom otor. Instructional K now ledge has been used in the field of ITS for approxim atel y a quarter of a century, and experience gained in the building of ITSs show s that they are often curriculum or topic oriented; learner m odel i ng i s oriented tow ard control; existing instructi onal know ledge is som etim es used m ore to serve technical design needs rather than learning needs. R ecent efforts in the A IE D com m unity appeared tow ard ITS-A uthoring [START_REF] Authier | A n authoring environm ent for curriculum and course building i n an ITS[END_REF][START_REF]Special Purpose O ntologies and the R epresentation of Pedagogical K now ledge[END_REF][START_REF]A n ITS authoring tool: E xperim ental advanced instructional advisor[END_REF][START_REF] Ikeda | Task O ntol ogy M akes It E asi er To U se A uthoring Tools[END_REF]. M urray [START_REF]A ut horing K now l edge-Based Tut ors: Tool s for Cont ent , Inst ruct i onal St rategy, St udent M odel , and Int erface D esi gn[END_REF] indi cates trends tow ard inclusi on, if not integration, of four com ponents : Tools for C ontent, Instructional Strategy, Student M odel, and Interface D esign. Intelligent A uthoring E nvironm ents that can support the building of ITSs need foundations in Instructional Science, w ith a coherent set of concepts and principles for building quality products. Such environm ents should provide authors w ith a choice betw een long established know ledge and m ore recent developm ents, such as R eigeluth s proposal to consider learners as co-designers of their instructi on, w here learners have the capability to request the com puter system to use som e instructional strategies, as w ell as the com puter deciding on som e strategies based on learner input [START_REF] Rei Gelut H | I s t he New Paradi gm of I nst ruct i onal Theory[END_REF]. A n IDaw are A uthoring System w ould know the distinction betw een designing an IIS, an Interactive Learning E nvironm ent (ILE ), and an O pen Learning E nvi ronm net (O LE ). It w ould provide the requirem ents and decisions to be m ade i n each case before starting any authoring, in order to have a com plete, coherent and congruent product. R equirem ents in designing an IIS rely on the know ledge of student and context as m uch as of the didactic know ledge. E xpl icit statem ents w ould be to specify the conditions of learning for w hi ch the system has been thought, as: com plem ent, supplem ent or replacem ent of teaching. D esi gning an ILE requires a different set of decisions, that can refer to either individual or team -based learning, w ith a philosophy such as situated learning; having fundam entals for a constructivist design for exam ple, helps us in m aking explicit statem ents about the design principles used, the authoring decisions m ade, and about their pedagogical finality and effectiveness. D esigning an O LE contains challenges that seem to be particularly in phase w ith the spirit of the tim e as w e step into the XXIst century. B ei ng open can m ean keeping your eyes open, and also being open-m i nded. W hat does it m ean for an O LE ? R equirem ents for an O LE typically are: 1) to know about external learning events, both those planned and the ones that happened, 2) to be able to reason, m ake hypothesis and decisions based on both internal and external events, 3) to be flexible in adapting instructi onal strategies based on culture or affects.

E xpl orations in the direction of an O ntology of Instruction for ID -aw are authoring environm ents have been described. C onclusi ons are that an O ntology of Instruction w ould be beneficial to the developm ent of the field; it w ould also benefit the field of Instructional Science as it has the capacity to stim ul ate reform ulations and the building of taxonom ies, w hil e, at the sam e tim e, consider new ideas and paradigm s a richness. R eigeluth s claim for a new paradigm of Instructional Theory [START_REF] Rei Gelut H | I s t he New Paradi gm of I nst ruct i onal Theory[END_REF]

  contains keyw ords such as: custom isation, autonom y, co-operation, shared decision-m aki ng, ini tiative, diversity, netw orking, hol ism , process-oriented, and Learner as K ing !

	R R R R e e e ef f f fe e e er r r re e e en n n n c c c ce e e es s s s
	[1]	M i zoguchi,	R .	A	Step	tow ards	O ntol ogical E ngineering.