CONVERGENCE RESULTS OF A HETEROGENEOUS ASYNCHRONOUS NEWMARK TIME INTEGRATORS

. This paper is concerned with the convergence analysis of the PH heterogeneous asynchronous time integrators algorithm, proposed by Prakash and Hjelmstad [ Int. J. Numer. Methods Eng. 61 (2004) 2183–2204], and devoted to transient dynamic problems for structural analysis. According to PH method, the time discretization is performed using the well-known Newmark schemes, where the time step ratio, i


Introduction
The introduction of the mixed time integration methods with/without different time steps (commonly known as heterogeneous asynchronous time integrators) for dynamic problems [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] are of considerable interest to many applications involving subdomains decomposition techniques (wave propagation [11,17,18], fluid-structure interaction problems [19], non-smooth contact problems [14]).The principle of heterogeneous asynchronous time integrators (HATI), for transient dynamic problems, consists on splitting the global problem into several subdomains, each one integrated by its own time scheme and/or its own time step.The main advantage of these methods is to be able to integrate the whole problem with several times steps (with/without heterogeneous time schemes) instead of a single time step usually controlled by the smallest element in the mesh, especially for conditionally stable schemes.Among the most popular time schemes currently used for structural dynamic problems, can be cited Newmark schemes [20], -schemes [21][22][23] and the Runge-Kutta schemes [24].In this paper, we focus on the convergence analysis of the mixed time integration method developed by Prakash and Hjelmstad in [25] for structural dynamic problems where the time discretization is performed via the Newmark schemes family.
To clarify the purpose of the paper, let us consider Ω as the set of nodes in R  ( = 1, 2 or 3 in general), which represent, for instance, the nodes of a mesh or the set of discrete points like particles.Assume a partition E. ZAFATI of Ω into two parts Ω 1 and Ω 2 such that the intersection Ω 1 ∩ Ω 2 = Γ 12 represents the interface separating the two sub-domains (see Fig. 1a).In this paper, we are concerned with the following classical system of equations which describes the transient dynamic problem for the whole domain Ω = Ω 1 ∪ Ω 2 over the open interval (0,  ): 1 Ü1 () +  1 U1 () +  1  1 () +    1 () =  1 ()  2 Ü2 () +  2 U2 () +  2  2 () +    2 () =  2 ()  1 U1 () +  2 U2 () =   () U1 (0) =  01 and  1 (0) =  01 U2 (0) =  02 and  2 (0) =  02 (1.1) where the vector valued functions Ü , U and   , associated with the subdomain Ω  with  ∈ {1, 2}, stand for the acceleration, velocity and displacement vectors of length   , respectively, while the dot symbol stands for the time derivative.The vector-valued functions  →   () are the applied external forces with length   .The two last equations of the previous system represent the initial conditions for suitable real vectors  0 and  0 of length   .In the context of mechanical engineering, the vector-valued function  of length   play the role of the Lagrange multipliers (we conserve this definition in the paper) and ensure the continuity of the interface forces between the two subdomain.Moreover, we assume the following: (H1): (1)   is symmetric positive definite matrix for every  ∈ {1, 2}.
(3)   is symmetric positive semi-definite matrix for every  ∈ {1, 2}.(4) The real matrix Since the nodes located at the interface are duplicated (Fig. 1b), the continuity of the velocity at the interface is ensured via the third equation of the system (1.1).Therefore, the operator (︀  1  2 )︀ are constructed to ensure the continuity of the velocity at the interface Γ 12 but also the Dirichlet conditions imposed in terms of the velocity for either Ω 1 or Ω 2 .We assume that the Dirichlet conditions are represented by the vector-valued function  →   () mapping (0,  ) to R   which includes the zero components corresponding to the continuities at the interface and satisfies, in addition, the following compatibility condition: In the context of this paper, the nodes at the interface does not necessary coincide (Fig. 2) which is of interest to the problems with non-conformal meshes.However, the only condition we should deal with is the surjectivity of (︀  1  2 )︀ which is a necessary for the well-posedness (see for instance Lem.2.2 in [26]).The aim of this paper is the analysis of the stability and the convergence of the PH method applied to the system (1.1) taking into account the assumption (H1).According to PH method, the numerical solution is computed at each macro time step by solving a generalized saddle-point problem [26], where the unknowns are the kinematic quantities, related to both subdomains, as well as the Lagrange multipliers.Since the Lagrange multipliers are only computed at the macro time scales, their approximations on the fine scale is performed by mean of a linear interpolation.Under some specific regularity conditions on external loads, we shall prove that the approximated solutions converge uniformly, with respect to the norm L ∞ , to the exact solution of the problem.We shall also establish some error estimates under a particular constraint on the Newmark parameters which is often satisfied in practice.To the best of the author's knowledge, a rigorous study on the convergence of the PH method under the assumption (H1) is not available at present.It is important to underline that the energy stability (see Lem. 3.5 below) has been established for the PH method in [25].However, it should be stressed that the latter definition of stability, in the context of PH method, is not generally sufficient to ensure the global boundedness of the set of solutions.Indeed, the investigation of the non-singularity of the global matrix generated by the PH method in [26] has shown that the existence and uniqueness of the numerical solution may fail if the matrix  1 is only positive semi-definite and  2 is not onto, i.e., an example of illustration has been given in Remark 3.9 in [26] with numerical validations.In this case, the set of the approximated solutions, assumed not empty, necessary contains an affine subspace.Thus, the set of solutions is unbounded despite the fact that the energy stability is satisfied.In this paper, we shall proceed differently and more rigorously to study the global stability and the convergence of the PH method under the hypothesis (H1).
The paper is organized as follows: a brief description of the PH algorithm is given in Section 2. In Section 3, we shall introduce the definitions of the approximated solutions and establish the boundedness results of the numerical solutions.Finally, the last section, i.e., Section 4, will be devoted to the discussion of the main results on the convergence of the PH method.

Review of PH method
The application of the PH algorithm on the system (1.1) is performed as follows: First, we choose the micro time scale ℎ 1 for the subdomain Ω 1 and the coarse time scale ℎ 2 for the subdomain Ω 2 linked by ℎ 2 = ℎ 1 , where  ∈ N * is a positive integer called the time step ratio.The equation of motion for the subdomain Ω 1 is described at the time (1)  = ℎ 1 ( = 0, 1, 2, . ..), while the equation of motion of the subdomain Ω 2 is described at the time for each  ∈ {0, 1, 2, . ..}.Now, let Ü 1 , U 1 ,   1 ,   1 and   1 be the acceleration, the velocity, the displacement, the external force and the Lagrange multiplier vectors, respectively, related to the subdomain Ω 1 at the time 2 and   2 be the acceleration, the velocity, the displacement, the external force and the Lagrange multiplier vectors, respectively, related to the subdomain Ω 2 at the time (2)  = ℎ 2 = ℎ 1 .The time discretization of the system using the GC method within the range [︁ (2) ]︁ ( ∈ N * ) is described as follows: For a fixed  ∈ (( − 1), ], the equilibrium equation for the subdomain Σ 1 at the time (1)  writes: The approximations of the quantities   1 and U 1 using a Newmark scheme, in terms of the parameters  1 and  1 , read: The equilibrium equation for the subdomain Σ 2 at the time writes: Similarly, the approximations of the quantities   2 and U 2 using a Newmark scheme, characterized by the parameters  2 and  2 , are given by: (2.4) The computation of the Lagrange multiplier at the fine scale, between two coarse time instants ( − 1)ℎ 2 and ℎ 2 , as proposed in the PH method is equivalent to the following interpolation: where In this paper, the term   2 is assumed to be zero which occurs, for instance, when no external loads are applied to the interface Γ 12 , i.e.,  2   2 = 0 for every index , or when the external force  2 are linear or constant with time.Thus, equation (2.5) implies: (2.7) Finally, the remaining equation concerns the continuity of the velocities at the interface as well as the Dirichlet conditions which are only imposed at the coarse time steps.More precisely, where    is the value of   at the time .The computation of the different quantities in (2.2)-(2.7)assumes the knowledge of the initial values  0 1 =  0 2 , Ü0 1 and Ü0 2 which are arbitrary in the context of this paper.The only requirement is that the initial values satisfy the equilibrium equations as described in (2.1) and (2.3) which has a meaning from a physical point of view.In practice, if   is sufficiently smooth, these initial values are generally computed using the following system: (2.9) It is more convenient to rewrite the previous time discretized equations in a block matrix representation as proposed in [25]: . . .
where for each  ∈ {1, 2} and each integer  ∈ [( − 1) + 1, ] for fixed , we have: Furthermore, one may notice that the system (2.10) can be cast in the following form: ]︂ (2.12) The previous system (2.12) is referred as the global problem in the paper and the involved matrix as the global operator.Comparing equations (2.12) and (2.10), the representation is unique and it is clear and the different operators A, L and B are given by: It is important to emphasize that the previous problem (2.12) may be viewed as a particular case of a generalized saddle-point problem and it is worth noting that similar system appears in other multi-time steps methods involving other time schemes (see for instance [10,12]).The generalized saddle-point problems have been investigated by Nicolaides [27] in the context of Hilbert spaces using the inf-sup conditions and then by Bernardi et al. [28] (see Thm. 2.1 in [28]) by extending the study to Banach spaces using Brezzi's assumptions (see Thm. 1.1 in [29]).In the context of PH method, The non-singularity of the global matrix in (2.12) as well as the strict-positivity of the Schur complement have been studied in [26] depending on the surjectivity of the link matrix  2 and using a special case of damping matrices (Rayleigh damping).Following the references [25,26], we consider the following assumptions on the Newmark parameters: (H2): For every  ∈ {1, 2}, the triple (  ,   , ℎ  ) satisfies: (1)

Boundedness of the numerical solution
In this section, we introduce some definitions of the approximated numerical solutions considered in the paper.We shall also discuss the existence and the stability of the numerical solutions under the assumptions (H1), (H2) and (HF) (Hypothesis (HF) is introduced below).
Let us first briefly recall some definitions of classical functional spaces.For every open  ⊂ R, we consider the following spaces: the space of all Lebesgue measurable functions  :  ↦ → R  such that the th derivative  () is square integrable for every  ∈ 0, . . ., ,  () ∈ L 2 (︀ , R  )︀ .The previous space is equipped with norm -If  is either open or closed the space C  (︀ , R  )︀ is the set of vector-valued functions, together with their classical derivatives of order lower than or equal to , continuous on .The Banach space C  (︀ For the sake of conciseness, the definitions of the previous norms are independent of the dimension . Let  = (0,  ) be the open interval and ℎ a positive reel such that  = ℎ, where  ≥ 1 is a positive integer.Let (︀   )︀ 0≤≤ be a finite sequence of vectors that represent the values of the quantity  computed at the times  0 = 0,  1 = ℎ,  2 = 2ℎ, . . .,   = ℎ.To every vector (︀   )︀ 0≤≤ we associate the measurable functions Xℎ, and Xℎ defined on  by: The definition (3.1) together with (3.2) imply that Xℎ is continuous over Ī.Moreover, if  = 1, Xℎ, is simply denoted Xℎ .
Remark 3.1.As a rule, the (weak) time derivative of Xℎ is denoted Ẋℎ and should not be confused with the dot notation on the discrete vector ( Ẋ )  which refers to the discrete quantities related to the derivative of .

For instance, Uℎ2
2 is the time derivative of the interpolation Ũℎ2 2 constructed from the sequence ( U 2 )  Given measurable functions  1 ,  2 and   as described in the problem (1.1), we say that the triple ( 1 ,  2 , ) of measurable functions in Ī is a solution of the problem (1.1) if  1 and  2 are twice weakly differentiable and satisfy (1.1) almost everywhere in  together with the initial conditions.The next result discusses the existence and the uniqueness of the solution of system (1.1) in the continuous framework.We shall state it under general assumptions: Theorem 3.2 (Existence and uniqueness).For every  ∈ {1, 2}, consider the hypothesis (H1) but with assumptions 2 and 3 replaced by: (1)   : R  ↦ → R  is single-valued and maximal monotone.
(2)   : R  ↦ → R  is continuous (not necessary linear) and exact, i.e., there exists some Gateau-differentiable function   : R  ↦ → R such that   =   .Moreover, we assume that   is nonnegative and   satisfies the following growth: where   is a constant independent of .
If the maps  →   () are measurable functions with . In the special case where   are linear symmetric and positive semi-definite, the solution is unique.

E. ZAFATI
Proof.See Appendix A.
In the following, we consider the following assumptions on the external loads: (HF): (1) For every  ∈ {1, 2}: In the sequel, it is not restrictive to consider by the continuous representative argument.In this case, it is seen from the proof of Theorem 3.2 that (HF) implies that Ü1 , Ü2 and  are continuous on .Since the initial problem (1.1) may be reduced to a problem with zero Dirichlet conditions, we will assume in the remainder of the paper that   = 0. Indeed, using the condition (HF) (2), we can replace  1 and  2 by  1 −  1 and  2 −  2 , respectively, where the couple for  = 1, 2, and satisfies  1 U1 () +  2 U2 () =   () for every  ∈ Ī.In this case, the right hand sides of the two first equations in (1.1) will contains further terms, each one belongs to W 1,2 (︀ , R  )︀ ,  = 1 or 2 (note that the existence of ( 1 ,  2 ) is guaranteed by the surjectivity of the operator ).For reading convenience, the condition "  = 0" will be systematically recalled.
The two next lemma describe some basic facts on the PH method, necessary to prove the global boundedness of the numerical solutions in Lemma 3.5.The reader may also refer to [26] for more details.Lemma 3.3.Assume that  =  ℎ2 ≥ 1 is a positive integer and denote ⟨•, •⟩ the duality bracket of an Euclidean space R  (the notation is independent of the dimension for simplification).Under the hypothesis (H1), (H2) and (HF) with   = 0, assume that the system (2.12) has a solution for every 1 ≤  ≤ , we claim: where and, Proof.This is too similar to the proof given in [25] see also [26].
Lemma 3.4.The matrix A is non-singular and the (, )-th block matrix entry of the inverse of A is given by: Proof.It is enough to check that the multiplication of the block matrix (3.7) with A gives the identity matrix.
The "energy stability" result as a consequence of Lemma 3.3 has been originally proved in [25], in the case of zero damping  1 =  2 = 0, which implies, under the hypothesis of zero external loads, the uniform boundedness of some quantities at the macro-time steps.This result does not evidently ensures the uniform boundedness of all quantities as being highlighted in [26], where it has been proved that the global matrix in equation (2.12) may be singular when the damping matrix  1 is only positive semi-definite.Lemma 3.5 below provides a more complete proof of stability in the case of positive definite damping matrices and under sufficiently smooth external loads.Lemma 3.5 (Boundedness result).Under the hypothesis (H1), (H2)-( 1) and (HF) with   = 0, there exist some constants ℎ 10 , ℎ 20 > 0 for fixed  (with ℎ 20 = ℎ 10 ), such that if we define the set  ℎ by Then, for fixed (ℎ 1 , ℎ 2 ) ∈  ℎ , the problem (2.10) has one unique solution In particular: where ] and the coefficient   > 0 only depends on the constants   ,   , ℎ 10 , ℎ 20 , ,  and the matrices   ,   ,   and   ( ∈ {1, 2}).The same results hold also for Ũℎ  , Ũℎ  , Ũℎ  and λℎ  .Proof of Lemma 3.5.First let us fix ℎ 10 and ℎ 20 to be sufficiently small such that the triples ( 1 ,  1 , ℎ 10 ) and ( 2 ,  2 , ℎ 20 ) satisfy the assumption (H2) and we choose arbitrary (ℎ 1 , ℎ 2 ) ∈  ℎ .Moreover, we keep the definitions of  1 and  2 as in the definition of  ℎ (3.8).When no confusion should arise and in order to avoid useless repetitions, we shall denote   > 0 as an arbitrary constant, independent of ℎ 1 , ℎ 2 , and only depends on the constants   ,   , ℎ 10 , ℎ 20 , ,  and the matrices   ,   ,   and   ( ∈ {1, 2}).
First, assume that the system (2.12) has a solution for every 1 ≤  ≤  2 and for every ℎ 2 such that (ℎ 1 , ℎ 2 ) ∈  ℎ .For every  ∈ {1, 2}, let 0 <   ≤   be an integer and choose a constant  > 0. By virtue of Young's, Jensen's and Cauchy's inequalities, the terms  ext,  are bounded by: In the view of equation (3.4), the previous equation (3.13) and the definitions in equation (3.5), if  > 0 is strictly lower that the smallest eigenvalues of both  1 and  2 (these eigenvalues are strictly positive by assumption (H1)), we infer, for every 1 ≤  ≤  2 , that: Indeed, using equation (2.4), it is seen that the velocities ( U 2 ) 1≤≤2 are bounded by: By analogous arguments, the same result may be obtained for the displacement vector  2 .Now, it remains to prove the boundedness of the quantities at the micro time steps, associated with the subdomain Ω 1 , as well as the Lagrange multipliers.This can be achieved in several steps: (3.14) and the Cauchy-Schwarz inequality gives: Step 2.
is uniformly bounded, i.e., the displacement, the velocity and the acceleration vectors of the subdomain Ω 1 , computed at the macro time steps, are uniformly bounded.
Using assumption (H2) and equation (3.14), it is clear that the acceleration components computed at the macro time steps are uniformly bounded: To prove the same result for the displacement vector, pick up an integer 0 ≤  ≤  2 .By virtue of equation (2.2), we have: Thus, combining step 1 and equation (3.18), we end up with: As a consequence, we conclude: Step 3. Boundedness of the Lagrange multipliers: At each macro time step ℎ 2 , we have: where the equality   1 =   2 has been used (see Eq. (2.5)).Since (︀  1  2 )︀ is surjective and   ∈ W 1,2 (︀ , R  )︀ is bounded on , we have for every 1 ≤  ≤  2 : where the uniform boundedness of the quantities at the macro time steps has been used as well as the continuous embedding W Step 4. The kinematic quantities related to Ω 1 , computed at the micro time steps are uniformly bounded.
In the view of the explicit expression of A −1 in Lemma 3.4 and equation (2.10), the expression of the quantities at the fine time step related to the subdomain Ω 1 can be written, for every ( − 1)+1 ≤  ≤ , as: for 1 ≤  ≤ .Moreover, it is not difficult to prove that M −1 1 and N 1 are uniformly bounded when the step ℎ 1 varies in the compact [0, ℎ 10 ].Thus, combining steps 1-3 and again the continuous embedding W , one may conclude that: The uniform boundedness related to the L 2 -norm is simply a consequence of the continuous embedding for every integer .Now it remains to prove the existence and the uniqueness of the solution.For this purpose, it is sufficient to show that the null-space of the matrix 12) is zero.Indeed, taking into consideration the preceding boundedness results, one may infer that the null-space of A is necessary bounded, hence it is reduced to the single element {0}.This completes the proof.
Remark 3.6.If   ̸ = 0, we can still obtain the uniform boundedness of the numerical solutions, without transforming the problem as described in the paragraph before Lemma 3.5, but with the additional assumption:

Convergence results
This section is focused on the main results of the paper, namely Theorems 4.1, 4.3 and 4.4, which discuss the uniform convergence of the approximated numerical solutions to the solution of the main problem in equation (1.1) as well as the error estimates.For this purpose, we assume, throughout this section, that the hypotheses of Lemma 3.5 are satisfied with   = 0.Moreover, we choose a sequence (ℎ 1, , ℎ 2, ) ≥0 ∈  N ℎ converging to 0 (the set  ℎ is the same defined in Lem. which is the unique solution of the problem (1.1) in S.Moreover, we have: (2) U =   and V =   a.e.
Theorem 4.1 only shows that the convergence of the acceleration and the Lagrange multipliers are in the weak sense.However, one may notice that the L ∞ strong convergence of either the acceleration field or the Lagrange multipliers necessary imply the strong convergence of the second quantity with the same norm.Given the assumption of Theorem 4.1, we can still obtain a strong convergence using an alternative definition of the approximate solution, piecewise constant on each macro time interval of length ℎ 2, as follows: If  1 and  2 stand for one of the following fields: displacement, velocity, acceleration and the Lagrange multipliers, related to the subdomains Ω 1 and Ω 2 , respectively, we define the sequence of the piecewise constant functions ( Xℎ 2,

2
)  by: )︁ () )︁ () ( where, for every 1 ≤  ≤  2, , the averages X 1 and X 2 are defined by: (see the definitions in Eq. (3.1)).The following Theorem states a strong convergence of the acceleration and the Lagrange multipliers vectors, For every sequence (ℎ 1, , ℎ 2, ) ≥0 ∈  N ℎ converging to 0. In particular the previous result holds for every  ≥ 1 when  1 =  2 = 1 2 .Note that the particular choice  1 =  2 = 1 2 is usually used in practice since it does not introduce artificial damping to the solution.We end this part by the following theorem which establishes some error estimates of the PH method with respect to the macro time step.
Remark 4.5.It is interesting to point out that numerical investigations of the convergence of PH method have been conducted on split-oscillators (without damping) in [12], which show that the order of convergence is nearly preserved even with time step ratios close to 20.Now, we shall give seperatly the proofs of the previous results.

Proof of Theorem 4.1
The proof is divided into several steps: By the continuity of  and the uniform boundedness of ( λℎ 1, 1 )  and ( λℎ 2, 2 )  , it is not difficult to observe that: Hence,  1 =  2 (a.e.).
Claim 3.For every  ∈ {1, 2},   and U may be selected to be continuous over Ī with   (0) =  0 and U (0) =  0 and up to a subsequence, we claim: = 0 and lim Furthermore, we have: = 0 and lim Henceforth,   and U may be assumed to be continuous on Ī.
Proof of Claim 3. We choose  = 1 or 2 and we restrict ourselves with the first convergences in equations ( 4 where  is independent of ℎ , .In equation ( 4 By Claim 3, and using the same subsequence, we also have: where the weak convergences are in the L 2 sense.Moreover, one may notice that:  (4.19) shows that the constraint relation in equation (1.1) is also satisfied, namely,  1 U1 + 2 U2 = 0.The convergence of the terms related to the external forces is a direct consequence of the regularity assumed in (HF).
As a consequence, it follows that the triple ( Finally, by the uniqueness of the solution according to Theorem 3.2, one may infer that the claims of Theorem 4.1 are satisfied for every sequence (ℎ 1, , ℎ 2, ) ≥0 ∈  N ℎ converging to 0, in particular, the convergences in (4.2) and (4.1).This completes the proof.

Proof of Theorem 4.3
Using the second relation in equation (2.2), we have for every integer 0 ≤  <  2, : and

.24)
Taking into account the compatibility condition (1.2) and the relation (2.8), we have for every integer 0 ≤  ≤  2, : Thus, for every integer 0 ≤  <  2, : Using the sequence (ℎ 2, )  , we infer: Moreover, using the linearity of the problem (1.1), we end up with: () = 0. (4.28) The key of the proof is the following equality: Equation (4.29) is a direct consequence of elementary computations taking into account the assumption  1 =  2 − −1 2 as stated in Theorem 4.3.By the uniform convergence in Theorem 4.1 and the regularity of external loads in assumption (HF), we have: Indeed, restricting our focus on the displacements, the triangular inequality entails: Moreover, using Lemma 3.5 and equation (2.2), we write for  = 1 and on each micro interval  , = [ + ,  ++1 ) (0 ≤  <  and  <  1, ): where   is independent of ,  and ℎ 2, .Following the same argument, the same may be obtained for  = 2. Thus: lim The same reasoning may be used to prove the remaining equations in (4.30).Now, by virtue of equations (4.28) and (4.29), we have for every  ∈ : Combining the strict non-negativity of the mass matrices and the Cauchy-Schwartz inequality, we get: where  1 is independent of ℎ 2, .As a consequence, using (4.30), we deduce: = 0 and lim By the linearity of the system (4.28),we get the same result for the Lagrange multipliers and this completes the proof.

Proof of Theorem 4.4:
If  = 1 or 2, we define the piecewise twice continuously differentiable function  ∼ ℎ ,  by: )︁ () In this proof, we shall frequently use the notation  to mean that the equality  = ((ℎ 2, )), where  is a vector possibly depends on the time  ∈  and the scalar  depends on ℎ 2, , is equivalent to the existence of a constant  independent of the time step ℎ 2, and  ∈  such that ‖‖ ≤ (ℎ 2, ) for all .In order to establish the estimates (4.6), it is sufficient to prove, for every 0 ≤  ≤  , that: where the time dependant quantities   satisfy   (ℎ 2, ) = (ℎ 2, ).The maps   and   are continuous, mapping Ī to the space of square matrices and independent of the time step ℎ 2, .Notice that the conclusions of the Theorem follow if equation (4.38) holds.Indeed, a direct application of Jones inequality [31] (see also Thm. 1.2.1 in [32]), a generalized form of the Gronwall inequality for piecewise continuous functions, shows that for  = 1, 2: Moreover, using equations (2.2), (2.4) and Lemma 3.5, it is not difficult to prove:  , we get the desired result for the displacement and the velocities vectors in equation (4.6).Moreover, using equations (4.35) and (4.39) together with the following estimates: = (ℎ 2, ).(4.41) The estimates for the acceleration and the Lagrange multipliers vectors follow immediately, where the Lagrange multipliers are expressed directly using equation (4.28).The proof of the first equality in (4.41) is simple.Indeed, restricting to  = 1, Ûℎ + , ++1 ), as: = (ℎ 2, ).The case  = 2 and the second equality in (4.41) follows from similar arguments.The third equality is a consequence of the regularity assumption on external loads.Now, we take only the two first formula in (4.38), the proof of the remaining ones being analogous.In the case of the displacement vector, it is sufficient to observe: where where ⌊⌋ is the greatest integer less than or equal to .This completes the proof for the displacements.Now, focusing on the velocities vectors and using the fact that Ü , we get: In the second equality of equation (4.45),only the equilibrium equation in (1.1) have been used as well as the definitions of the approximate solutions, i.e., equations (3.1) and (4.3), noticing that: Now, we shall estimate  1 ,  2 ,  3 and  4 .We start  1 by omitting the constant multiplicative terms: Moreover, using the boundedness results in Lemma 3.5, one may notice that for every : ++1 ) (0 ≤  <  and  <  2, ) as follows: )︁ Taking into account the previous equation ( 4.49), we write: )︁ d.(4.50) By the boundedness results in Lemma 3.5, we have: = 0, we have: Combining equations (4.49) to (4.52), we deduce that  3 = (ℎ 2, ).With the focus on the remaining term, i.e.,  4 , it is easy to see that  4 = (ℎ 2, ) using the strong assumptions on the regularity of external loads in Theorem 4.4.In fact, for every  ∈ [ +1 ]( ≤  1, ): where As previously, we show that   = (ℎ 2, ) ( ∈ {5, 6, 7, 8, 9, 10}).We start with  5 and we write by omitting the useless constant terms: )︁ d = (ℎ 2, ).Moreover, for  < ⌊  ℎ 2, ⌋, we have: where  2 (ℎ 2 ) = (ℎ 2, ).The functions  2 and  2 are continuous which map Ī to the space of square matrices.The proof is complete.

Conclusions
In this paper, the convergence analysis of the PH heterogeneous asynchronous Newark time integrators has been studied, i.e., more precisely in Theorems 4.1, 4.3 and 4.4, using suitable definitions of the approximate solutions.The analysis is mainly based on the uniform boundedness results of the numerical solutions established Lemma 3.5.It turns out that for sufficiently regular external loads, uniform convergence with respect to the norm L ∞ is achieved.Moreover, the estimates furnished in Theorem 4.4, under particular conditions on Newmark parameters and external loads, show that the error convergence is at least of the first order with respect to the macro time step.All the results in this work are obtained under the assumptions of damped domains and linear framework; it would be interesting to generalize these results by considering non-linearities or weakening the damping assumptions.

Figure 1 .
Figure 1.The whole domain Ω splitted into two subdomains Ω 1 and Ω 2 (a) where the nodes at the interface Γ 12 are duplicated (b).