Research article Special Issues

A new strict decay rate for systems of longitudinal m-nonlinear viscoelastic wave equations

  • Recent years have been marked by a significant increase in interest in solving nonlinear equations that arise in various fields of natural science. This trend is associated with the creation of a new method of mathematical physics. The present study is devoted to the analysis of the propagation of m-nonlinear viscoelastic waves equations in an unbounded domain. The physical properties are determined by the equations of the linear theory of viscoelasticity. This article shows the main effect and interaction between the different weak and strong damping terms on the behavior of solutions. We found, under a novel condition on the kernel functions, an energy decay rate by using an appropriate energy estimates.

    Citation: Keltoum Bouhali, Sulima Ahmed Zubair, Wiem Abedelmonem Salah Ben Khalifa, Najla ELzein AbuKaswi Osman, Khaled Zennir. A new strict decay rate for systems of longitudinal m-nonlinear viscoelastic wave equations[J]. AIMS Mathematics, 2023, 8(1): 962-976. doi: 10.3934/math.2023046

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  • Recent years have been marked by a significant increase in interest in solving nonlinear equations that arise in various fields of natural science. This trend is associated with the creation of a new method of mathematical physics. The present study is devoted to the analysis of the propagation of m-nonlinear viscoelastic waves equations in an unbounded domain. The physical properties are determined by the equations of the linear theory of viscoelasticity. This article shows the main effect and interaction between the different weak and strong damping terms on the behavior of solutions. We found, under a novel condition on the kernel functions, an energy decay rate by using an appropriate energy estimates.



    This paper is concerned with the finite element approximation of system of J = 2 quasi-variational inequalities QVIs with term sources and obstacles depending on solution: Find a vector U=(u1,u2)(H10(Ω))2 satisfying

    ai(ui,vui)(fi(ui),vui);vH10(Ω) (1.1)
    v,uiMui;ui0.

    Where Ω is a bounded smooth domain of RN with N 1, each ai(.,.) is a continuous elliptic bilinear form, (.,.) is the inner product in L2(Ω) and each fi is a regular, nonlinear functional depending on solutions. The obstacle M provide the coupling between the unknowns u1; u2

    Mui=k+infμiuμ;

    k is a positive number. We point out that in the case where fi are independent of the solution, the system (1.1) coincides with that introduced by Bensoussan and Lions in [1] which arises in the management of energy production problems.

    It is easy to note that the structure of system (1.1) is analogous to that of the classical obstacle problem [2] where the term source and obstacle are depending upon the solution sought. The terminology QVI being chosen is a result of this remark.

    Numerical analysis of system of quasi-variational inequalities where term sources not depending on solutions were achieved in several works, we refer to [3,4,5,6,7,8] for system of quasi-variational inequalities with coercive or noncoercive operators.

    For results on systems related to evolutionary Hamilton-Jacobi-Bellman equation we refer to [9,10,11].

    The main objective of this paper is to show that problem (1.1) can be properly approximated by a finite element method and an optimal L-error estimates is derived, which coincides with the optimal convergence order of elliptic variational inequalities of an obstacle type problem [12].

    The approximation is carried out by first introducing a modified Bensoussan-Lions type iterative scheme depending on parameters which is shown to converge geometrically to the continuous solution. By a symmetrical approach, using the standard finite element method and a discrete maximum principle (DMP), the geometric convergence of the discrete modified Bensoussan-Lions type iterative scheme depending upon parameters is given as well. An L-error estimates is then established combining the geometric convergence of both the continuous and discrete iterative schemes and the known uniform error estimates in elliptic VIs.

    It is worth mentioning that even the guiding idea of this paper rests on the algorithmic approach followed in many papers cited above, the treatment of the geometric convergence of both continuous and discrete schemes is totally different because of the nonlinear nature of terms sources. Also, it is used for the first time for a system of QVIs.

    An outline of this paper is as follows: In section 2, we lay down some definitions and classical results related to variational inequalities and prove a Lipschitz continuous and discrete dependency with respect to the source term, the boundary condition and the obstacle. Section 3 discusses the continuous Bensoussan-Lions type iterative scheme and proves its geometrical convergence. In Section 4, we establish the finite element counter parts of the continuous system and the continuous Bensoussan-Lions type iterative scheme respectively and the geometrical convergence of the discrete scheme. Section 5 is devoted the L-error analysis of the method.

    We are given functions aijk(x),aik(x),ai0(x),1i2 sufficiently smooth functions such that 1j,kN

    1j,kNaijk(x)ξjξkα|ξ|2,ξRN,α>0
    ai0(x)βi>0,(xΩ) (2.1)

    where βi is a positive constant. We define the bilinear forms: For all u,vH10(Ω)

    ai(u,v)=Ω(1j,kNaijk(x)uxjvxk+Nk=1aik(x)uxkv+ai0(x)uv)dx (2.2)

    We are given right-hand sides

    fisuchthatfiL(Ω),fif0>0,

    a nonlinear functional and Lipschitz continuous on R; that is

    |fi(x)fi(y)|ki|xy|,x,yR

    such that

    αi=kiβi<1, (2.3)

    where βi is a constant defined in (2.1). For W=(w1,w2)(L+(Ω))2 we introduce the norm

    W=max1i2wiL(Ω).

    Let be Ω a bounded polyhedral domain of R2 or R3 with sufficiently smooth boundary Ω. We consider the bilinear form of the same form of those defined in (2.2), the linear form

    (f,v)=Ωf(x)v(x)dx (2.4)

    The right hand side

    fL(Ω), (2.5)

    the obstacle

    ψW2,(Ω)andψ0, (2.6)

    the boundary condition gL(Ω) and the nonempty convex set

    Kg={vH1(Ω)suchthatv=gonΩandvψonΩ}. (2.7)

    We consider the variational inequality V.I.: Find uKg such that

    a(u,vu)(f,vu),vKg. (2.8)

    Proposition 1 Let (f,g,ψ); (˜f,˜g,˜ψ)be a pair of data and ζ=σ(f,g,ψ); ˜ζ=σ(˜f,˜g,˜ψ) the corresponding solution to (2.8). If f˜f in Ω, g˜g on ∂Ω and ψ˜ψ then, ζ˜ζ in Ω.

    Proof. The proof is an adaptation of the proof of the monotonicity property of the solution of Ⅵ with nonlinear source term (see [13]). According to [14], ζ=max{ζ_} where {ζ_} is the set of all the subsolutions of ζ. Hence, ζ_{ζ_}, ζ_ satisfies

    a(ζ_,v)(f,v),v0withζ_ψandζ_g.

    By using the conditions f˜f in Ω, g˜g on ∂Ω and ψ˜ψ, we get

    a(ζ_,v)(f,v)(˜f,v),

    with

    ζ_ψ˜ψandζ_g˜gonΩ.

    Thus, ζ is a subsolution of ˜ζ=σ(˜f,˜g,˜ψ), that is ζ˜ζ in Ω.

    This subsection is devoted to the establishment of a Lipschitz continuous dependence property of the solution with respect to the source term, the boundary condition and the obstacle by which we first, set out and demonstrate.

    Proposition 2 Let (f,g,ψ); (˜f,˜g,˜ψ)be a pair of data andζ=σ(f,g,ψ); ˜ζ=σ(˜f,˜g,˜ψ) the corresponding solution to (2.8). Then, we have

    ζ˜ζL(Ω)max{(1β)f˜fL(Ω),g˜gL(Ω),ψ˜ψL(Ω)}. (2.9)

    Proof. The proof is an adaptation of the proof of a Lipschitz property of the solution of Ⅵ with nonlinear source term (see [13]). First, set

    φ=max{(1β)f˜fL(Ω),g˜gL(Ω),ψ˜ψL(Ω)}. (2.10)

    Then,

    ˜ff+f˜fL(Ω)
    f+(1)f˜fL(Ω)
    f+(a0(x)β)f˜fL(Ω)
    f+a0(x)max{(1β)f˜fL(Ω),g˜gL(Ω),ψ˜ψL(Ω)}.

    So,

    f+a0(x)φinΩ. (2.11)

    Thus, for all 0<v,

    (˜f,v)(f+a0(x)φ,v),

    with

    ˜ζ˜gg+φonΩ,
    ˜ζ˜ψψ+φinΩ.

    So, according to the property ˜ζ is a subsolution ofσ(f+a0(x)φ,g+φ,ψ+φ)=σ(f,g,ψ)+φ, that is

    ˜ζζ+φin¯Ω

    or

    ˜ζζφin¯Ω. (2.12)

    Similarly, interchanging the roles of the couples (f,g,ψ); (˜f,˜g,˜ψ), we obtain

    ζ˜ζφin¯Ω, (2.13)

    which completes the proof.

    Let τh be a triangulation of Ω with meshsize h, Vh be the space of finite elements consisting of continuous piecewise linear functions v vanishing on ∂Ω and φs; s = 1, 2, …, m(h) be the basis functions of Vh.

    The discrete counterpart of (2.8) consists of finding uhKgh such that

    a(uh,vuh)(f,vuh),vKgh. (2.14)

    Where

    Kgh={vVhsuchthatv=πhgonΩandvrhψonΩ}, (2.15)

    πh is an interpolation operator on Ω and rhis the usual finite element restriction operator on Ω.

    Theorem 3 (See [12] Under conditions (2.5) and (2.6), there exists a constant C independent of h such that

    ζζhL(Ω)Ch2|logh|2. (2.16)

    Assuming that the DMP is satisfied, i.e. the matrix resulting from the finite element discretization is an M-matrix (see [15,16]), we prove the Lipschitz discrete dependence with respect to the boundary condition, the source term and the obstacle by a similar study to that undertaken previously for the Lipschitz continuous dependence property.

    Proposition 4 Let (f,g,rhψ); (˜f,˜g,rh˜ψ)be a pair of data and ζh=σh(f,g,rhψ); ˜ζh=σh(˜f,˜g,rh˜ψ) the corresponding solution to (2.14). If f˜f in Ω, g˜g on ∂Ω and rhψrh˜ψ then, ζh˜ζh in Ω.

    Proof. The proof is similar to that of the continuous case.

    The proposition below establishes a Lipschitz discrete dependence of the solution with respect to the data.

    Proposition 5 Let the (d.m.p) holds. Then, we have

    ζh˜ζhL(Ω)max{(1β)f˜fL(Ω),g˜gL(Ω),rhψrh˜ψL(Ω)} (2.17)

    Proof. The proof is similar to that of the continuous case.

    We define the following fixed-point mapping

    T:(L+(Ω))2(L+(Ω))2
    Z=(z1,z2)TZ=ζ=(ζ1,ζ2).

    Where ζiH10(Ω)L(Ω) is a solution to the following variational inequality

    ai(ζi,vζi)(fi(zi),vζi);vH10(Ω) (3.1)
    v,ζiMζi=k+zj;ζi0withij.

    Thanks to [1,2], ζi is the unique solution to coercive variational inequality (3.1).

    Remark 1 We remark that the solution U=(u1,u2) of the system (1.1) is the fixed point of the mapping T; that isTU=U.

    Starting from U0=(u1,0,u2,0) where ui,0; i = 1; 2 is solution of the variational equation

    ai(ui,0,v)=(fi(ui,0),v),vH10(Ω),

    and for all 0<wi<1; i=1,2 we define the sequences (u1,n+1) and (u2,n+1) such that u1,n+1 and u2,n+1 the components of the vector Un+1, solve the following elliptic variational inequalities respectively

    (u1,n+1,vu1,n+1)(w1f1(u1,n+1)+(1w1)f1(u1,n),vu1,n+1) (3.2)
    v,u1,n+1Mu1,n+1=k+u2,n, (3.3)
    a2(u2,n+1,vu2,n+1)(w2f2(u2,n+1)+(1w2)f2(u2,n),vu2,n+1) (3.4)
    v,u2,n+1Mu2,n+1=k+u1,n+1. (3.5)

    Theorem 2 The sequences (u1,n+1) and (u2,n+1) converge geometrically to the solution U=(u1,u2) of the system (1.1); there exist a positive real ρ(0,1) which depends on αi and wi such that for all n0

    Un+1Uρn+1U0U (3.6)

    where

    ρ=max1i2α1(1w1)1α1w1<1. (3.7)

    Proof. The proof will carry out by induction.

    ● We first deal with the case

    u1u1,0L(Ω)=max1i2uiui,0L(Ω). (3.8)

    ● Indeed for n = 0; using (1.1), (3.2), (3.3) and (2.9), we have

    u1u1,1L(Ω)max{(1β1)f1(u1)(w1f1(u1,1)+(1w1)f1(u1,0))L(Ω);u2u2,0L(Ω)}
    max{(1β1)w1(f1(u1)f1(u1,1))+(1w1)(f1(u1)f1(u1,0))L(Ω);u2u2,0L(Ω)}
    max{(k1β1)(w1u1u1,1L(Ω)+(1w1)u1u1,0L(Ω));u2u2,0L(Ω)}.

    So,

    u1u1,1L(Ω)max{α1w1u1u1,1L(Ω)+α1(1w1)u1u1,0L(Ω);u2u2,0L(Ω)} (3.9)

    We distinguish two cases

    max{α1w1u1u1,1L(Ω)+α1(1w1)u1u1,0L(Ω);u2u2,0L(Ω)}
    =α1w1u1u1,1L(Ω)+α1(1w1)u1u1,0L(Ω) (3.10)

    or

    max{α1w1u1u1,1L(Ω)+α1(1w1)u1u1,0L(Ω);u2u2,0L(Ω)}
    =u2u2,0L(Ω) (3.11)

    (3.9) in conjunction with case (3.10) implies

    u1u1,1L(Ω)α1w1u1u1,1L(Ω)+α1(1w1)u1u1,0L(Ω) (3.12)

    with

    u2u2,0L(Ω)α1w1u1u1,1L(Ω)+α1(1w1)u1u1,0L(Ω) (3.13)

    which implies

    u1u1,1L(Ω)α1(1w1)1α1w1u1u1,0L(Ω). (3.14)

    By replacing (3.14) in (3.13), we get

    u2u2,0L(Ω)α1(1w1)1α1w1u1u1,0L(Ω)
    ρmax1i2uiui,0L(Ω),

    which coincides with (3.8).

    (3.9) in conjunction with (3.11) implies

    u1u1,1L(Ω)u2u2,0L(Ω) (3.15)

    with

    α1w1u1u1,1L(Ω)+α1(1w1)u1u1,0L(Ω)u2u2,0L(Ω). (3.16)

    u2u2,0L(Ω) is bounded below by both u1u1,1L(Ω)

    and

    α1w1u1u1,1L(Ω)+α1(1w1)u1u1,0L(Ω).

    So,

    u1u1,1L(Ω)α1w1u1u1,1L(Ω)+α1(1w1)u1u1,0L(Ω)

    or

    α1w1u1u1,1L(Ω)+α1(1w1)u1u1,0L(Ω)u1u1,1L(Ω).

    Then,

    u1u1,1L(Ω)α1(1w1)1α1w1u1u1,0L(Ω) (3.17)

    or

    α1(1w1)1α1w1u1u1,0L(Ω)u1u1,1L(Ω). (3.18)

    (3.15), (3.17) and (3.18) generate the following three possibilities

    u1u1,1L(Ω)α1(1w1)1α1w1u1u1,0L(Ω)u2u2,0L(Ω)max1i2uiui,0L(Ω)

    or

    u1u1,1L(Ω)u2u2,0L(Ω)α1(1w1)1α1w1u1u1,0L(Ω)max1i2uiui,0L(Ω)

    or

    α1(1w1)1α1w1u1u1,0L(Ω)u1u1,1L(Ω)u2u2,0L(Ω)max1i2uiui,0L(Ω).

    All possibilities are true in the same time because they coincide with (3.8). So, there is either a contradiction and thus case (3.11) is impossible or case (3.11) is possible if and only if

    u1u1,1L(Ω)=α1(1w1)1α1w1u1u1,0L(Ω).

    Hence, both cases (3.10) and (3.11) imply (3.14).

    ● Let us now discuss the second case

    u2u2,0L(Ω)=max1i2uiui,0L(Ω). (3.19)

    (3.9) in conjunction with (3.10) implies (3.14) with

    u2u2,0L(Ω)α1(1w1)1α1w1u1u1,0L(Ω)
    ρmax1i2uiui,0L(Ω)<u2u2,0L(Ω),

    which contradicts (3.19) which means that (3.10) is impossible. (3.9) in conjunction with (3.11) we get (3.17) and (3.18). So,

    u1u1,1L(Ω)α1(1w1)1α1w1u1u1,0L(Ω)max1i2uiui,0L(Ω)

    or

    α1(1w1)1α1w1u1u1,0L(Ω)u1u1,1L(Ω)max1i2uiui,0L(Ω).

    We remark that both alternatives are true in same time because both coincide with (3.19) which implies that in case (3.11), we must have

    u1u1,1L(Ω)=α1(1w1)1α1w1u1u1,0L(Ω).

    Hence, in both cases (3.8) and (3.19), we obtain (3.14). Hence,

    u1u1,1L(Ω)ρmax1i2uiui,0L(Ω). (3.20)

    ● As

    U1=(u1,1,u2,1)andU=(u1,u2),

    we need to deal also with u2u2,1L(Ω), by following the same reasoning as that adopted for u1 and u1,1, we get

    u2u2,1L(Ω)max{α2w2u2u2,1L(Ω)+α2(1w2)u2u2,0L(Ω);u1u1,1L(Ω)} (3.21)

    Again we distinguish two possibilities

    max{α2w2u2u2,1L(Ω)+α2(1w2)u2u2,0L(Ω);u1u1,1L(Ω)}
    =α2w2u2u2,1L(Ω)+α2(1w2)u2u2,0L(Ω); (3.22)

    or

    max{α2w2u2u2,1L(Ω)+α2(1w2)u2u2,0L(Ω);u1u1,1L(Ω)}
    =u1u1,1L(Ω). (3.23)

    (3.21) and (3.22) imply

    u2u2,1L(Ω)α2(1w2)(1α2w2)u2u2,0L(Ω) (3.24)

    with

    u1u1,1L(Ω)α2w2u2u2,1L(Ω)+α2(1w2)u2u2,0L(Ω). (3.25)

    By substituting (3.24) in (3.25), we get

    u1u1,1L(Ω)α2(1w2)1α2w2u2u2,0L(Ω)ρmax1i2uiui,0L(Ω),

    which coincides with (3.20). (3.21) and (3.23) imply

    u2u2,1L(Ω)u1u1,1L(Ω) (3.26)

    with

    α2w2u2u2,1L(Ω)+α2(1w2)u2u2,0L(Ω)u1u1,1L(Ω).

    It is clear that u1u1,1L(Ω) is bounded below by both

    u2u2,1L(Ω)

    and

    α2w2u2u2,1L(Ω)+α2(1w2)u2u2,0L(Ω)

    which leads us to distinguish the following possibilities

    u2u2,1L(Ω)α2w2u2u2,1L(Ω)+α2(1w2)u2u2,0L(Ω)

    or

    α2w2u2u2,1L(Ω)+α2(1w2)u2u2,0L(Ω)u2u2,1L(Ω).

    Then,

    u2u2,1L(Ω)α2(1w2)1α2w2u2u2,0L(Ω) (3.27)

    or

    α2(1w2)1α2w2u2u2,0L(Ω)u2u2,1L(Ω). (3.28)

    Thus, (3.26)-(3.28) imply that the three following alternatives are required

    u2u2,1L(Ω)u1u1,1L(Ω)α2(1w2)1α2w2u2u2,0L(Ω)

    or

    u2u2,1L(Ω)α2(1w2)1α2w2u2u2,0L(Ω)u1u1,1L(Ω)

    or

    α2(1w2)1α2w2u2u2,0L(Ω)u2u2,1L(Ω)u1u1,1L(Ω).

    It is clear that all alternatives coincide with (3.20). So, we must have

    u2u2,1L(Ω)=α2(1w2)1α2w2u2u2,0L(Ω).

    Thus, in both cases (3.22) and (3.23) we obtain (3.24). Hence,

    u2u2,1L(Ω)ρmax1i2uiui,0L(Ω). (3.29)

    (3.20) and (3.29) imply

    U1UρU0U.

    ● Let us assume that, for n0

    uiui,nL(Ω)ρnmax1i2uiui,0L(Ω),i=1,2. (3.30)

    ● We prove

    uiui,n+1L(Ω)ρn+1max1i2uiui,nL(Ω),i=1,2. (3.31)

    By adopting the same arguments for (1.1), (3.2), (3.3) and (2.9) as that applied for the previous iterates, we get

    u1u1,n+1L(Ω)max{(1β1)f1(u1)(w1f1(u1,n+1)+(1w1)f1(u1,n))L(Ω);u2u2,nL(Ω)}

    So,

    u1u1,n+1L(Ω)max{α1w1u1u1,n+1L(Ω)+α1(1w1)u1u1,nL(Ω);u2u2,nL(Ω)} (3.32)

    Also we distinguish two cases:

    max{α1w1u1u1,n+1L(Ω)+α1(1w1)u1u1,nL(Ω);u2u2,nL(Ω)}
    =α1w1u1u1,n+1L(Ω)+α1(1w1)u1u1,nL(Ω) (3.33)

    or

    max{α1w1u1u1,n+1L(Ω)+α1(1w1)u1u1,nL(Ω);u2u2,nL(Ω)}=u2u2,nL(Ω) (3.34)

    (3.32) in conjunction with (3.33) implies

    u1u1,n+1L(Ω)α1(1w1)1α1w1u1u1,nL(Ω), (3.35)

    with

    u2u2,nL(Ω)α1w1u1u1,n+1L(Ω)+α1(1w1)u1u1,nL(Ω). (3.36)

    By replacing (3.35) in (3.36) we get, according to (3.30); i = 1

    u2u2,nL(Ω)α1(1w1)1α1w1u1u1,nL(Ω)ρn+1max1i2uiui,0L(Ω)

    which matches with (3.30); i = 2. (3.32) in conjunction with (3.34) implies

    u1u1,n+1L(Ω)u2u2,nL(Ω) (3.37)

    with

    α1w1u1u1,n+1L(Ω)+α1(1w1)u1u1,nL(Ω)u2u2,nL(Ω).

    u2u2,nL(Ω) is bounded below by both u1u1,n+1L(Ω)

    and

    α1w1u1u1,n+1L(Ω)+α1(1w1)u1u1,nL(Ω)

    So,

    u1u1,n+1L(Ω)α1w1u1u1,n+1L(Ω)+α1(1w1)u1u1,nL(Ω)

    or

    α1w1u1u1,n+1L(Ω)+α1(1w1)u1u1,nL(Ω)u1u1,n+1L(Ω).

    Thus,

    u1u1,n+1L(Ω)α1(1w1)1α1w1u1u1,nL(Ω)

    or

    α1(1w1)1α1w1u1u1,nL(Ω)u1u1,n+1L(Ω).

    By taking into account (3.37), we get

    u1u1,n+1L(Ω)u2u2,nL(Ω)α1(1w1)1α1w1u1u1,nL(Ω)

    or

    u1u1,n+1L(Ω)α1(1w1)1α1w1u1u1,nL(Ω)u2u2,nL(Ω)

    or

    α1(1w1)1α1w1u1u1,nL(Ω)u1u1,n+1L(Ω)u2u2,nL(Ω).

    Three possibilities are true because all coincide with (3.30). So, we necessarily get

    u1u1,n+1L(Ω)α1(1w1)1α1w1u1u1,nL(Ω).

    Thus, both cases (3.33) and (3.34) imply (3.35). Hence, by using (3.30) we get (3.31) for i = 1. The proof for (3.31); i = 2 is obtain in similar way by using (3.31); i = 1 and (3.35) so, it will be omitted. The desired result (3.6) follows naturally from (3.31).

    This section, we will handle the discrete problem by a perfect symmetry in the treatment of that the continuous one. Indeed, we define the discrete system of QVIs: Find a vector Uh=(u1h,u2h)(Vh)2 such that

    {ai(uih,vuih)(fi(uih),vuih);vVhv,uihrh(Muih)=rh(k+ujh);ij.uih0anduih=πhgonΩ. (4.1)

    The related discrete fixed-point mapping

    Th:(Vh)2(Vh)2
    Zh=(z1h,z2h)ThZh=ζh=(ζ1h,ζ2h),

    where ζihVh is the unique solution to the following discrete variational inequality

    ai(ζih,vζih)(fi(zih),vζih);vVh (4.2)
    v,ζihrh(Mζih)=rh(k+zjh);ζih0withijandζih=πhgonΩ.

    Remark 1 We remark that the solution Uh=(u1h,u2h) of the system (4.1) is the fixed point of the mapping Th; that is ThUh=Uh.

    Starting from U0h=(u1,0h,u2,0h) where ui,0h=rhui,0;i=1,2 is the discrete analog of ui,0 then,

    ui,0ui,0hL(Ω)Ch2|logh|2. (4.3)

    For all 0<wi<1;i=1,2 we define the discrete sequences (u1,n+1h) and (u2,n+1h) such that u1,n+1h and u2,n+1h components of the vector Un+1h solve discrete elliptic variational inequalities

    a1(u1,n+1h,vu1,n+1h)(w1f1(u1,n+1h)+(1w1)f1(u1,nh),vu1,n+1h) (4.4)
    v,u1,n+1hrh(Mu1,n+1h)=rh(k+u2,nh), (4.5)
    a2(u2,n+1h,vu2,n+1h)(w2f2(u2,n+1h)+(1w2)f2(u2,nh),vu2,n+1h) (4.6)
    v,u2,n+1hrh(Mu2,n+1h)=rh(k+u1,n+1h). (4.7)

    Theorem 2 The discrete sequences (u1,n+1h) and (u2,n+1h) converge geometrically to the discrete solution Uh=(u1h,u2h) of the system (4.1); there exist a positive real ρ(0,1) defined in (3.7) such that for all n0

    Un+1hUhρn+1U0hUh. (4.8)

    Proof. The proof is similar to that of the continuous case.

    This section is devoted to the proof of the main result of this paper. For that purpose we need to introduce an auxiliary system.

    Let wi,0h=ui,0h;i=1,2 be an initialization. For all 0<wi<1;i=1,2 we define the discrete sequences (w1,n+1h) and (w2,n+1h) such that w1,n+1h and w2,n+1h solve coercive variational inequalities

    a1(w1,n+1h,vw1,n+1h)(w1f1(u1,n+1)+(1w1)f1(u1,n),vw1,n+1h) (5.1)
    v,w1,n+1hrh(Mu1,n+1)=rh(k+u2,n), (5.2)
    a2(w2,n+1h,vw2,n+1h)(w2f2(u2,n+1)+(1w2)f2(u2,n),vw2,n+1h) (5.3)
    v,w2,n+1hrh(Mu2,n+1)=rh(k+u1,n+1). (5.4)

    It is clear that wi,n+1h;i=1,2 components of the vector Wn+1h are finite element approximation of ui,n+1 defined in (3.2)–(3.4). Thus, making use of (2.16); we get

    wi,n+1hui,n+1L(Ω)Ch2|log|2;i=1,2andn0. (5.5)

    The algorithmic approach used in the present paper rests on the following crucial lemma, where the error estimate between the nth iterate Un and its discrete counter parts Un+1h is established.

    Lemma 1 Let (Un+1) and (Un+1h) be the vectors whose components are sequences defined in (3.2)–(3.5) and (4.4)–(4.7) respectively. Then,

    Un+1Un+1h(γ(1ρn+11ρ)+ρn+1)maxn0UnWnh. (5.6)

    Where

    γ=max1i2{1(1αiwi)}. (5.7)

    Proof. The proof of the lemma rests on the discrete Lipschitz continuous dependency with respect to source term and obstacle and will carry out by induction.

    ● For n = 0, we have

    u1,1u1,1hL(Ω)u1,1w1,1hL(Ω)+w1,1hu1,1hL(Ω).

    (5.1), (5.2), (4.4), (4.5) and (2.17) imply

    u1,1u1,1hL(Ω)u1,1w1,1hL(Ω)
    +max{(1β1)f1(u1,1)(w1f1(u1,1h)+(1w1)f1(u1,0h))L(Ω);rh(k+u2,0)rh(k+u2,0h)L(Ω)}

    So,

    u1,1u1,1hL(Ω)u1,1w1,1hL(Ω)
    +max{(k1β1)w1u1,1u1,1hL(Ω)+(k1β1)(1w1)u1,0u1,0hL(Ω);rh(k+u2,0)rh(k+u2,0h)L(Ω)}.

    Therefore,

    u1,1u1,1hL(Ω)u1,1w1,1hL(Ω) (5.8)
    +max{α1w1u1,1u1,1hL(Ω)+α1(1w1)u1,0u1,0hL(Ω);u2,0u2,0hL(Ω)}.

    We distinguish two cases

    max{α1w1u1,1u1,1hL(Ω)+α1(1w1)u1,0u1,0hL(Ω);u2,0u2,0hL(Ω)}
    =α1w1u1,1u1,1hL(Ω)+α1(1w1)u1,0u1,0hL(Ω) (5.9)

    or

    max{α1w1u1,1u1,1hL(Ω)+α1(1w1)u1,0u1,0hL(Ω);u2,0u2,0hL(Ω)}=u2,0u2,0hL(Ω) (5.10)

    (5.8) in conjunction with (5.9) imply

    u1,1u1,1hL(Ω)u1,1w1,1hL(Ω)+α1w1u1,1u1,1hL(Ω)
    +α1(1w1)u1,0u1,0hL(Ω)

    with

    u2,0u2,0hL(Ω)α1w1u1,1u1,1hL(Ω)+α1(1w1)u1,0u1,0hL(Ω). (5.11)

    So,

    (1α1w1)u1,1u1,1hL(Ω)u1,1w1,1hL(Ω)+α1(1w1)u1,0u1,0hL(Ω),

    with (5.11). Then,

    u1,1u1,1hL(Ω)1(1α1w1)u1,1w1,1hL(Ω)+α1(1w1)(1α1w1)u1,0u1,0hL(Ω). (5.12)

    By replacing (5.12) in (5.11) we obtain

    u2,0u2,0hL(Ω)α1w1(1α1w1)u1,1w1,1hL(Ω)+α1(1w1)(1α1w1)u1,0u1,0hL(Ω).

    According to (5.5) and (4.3) we get,

    u2,0u2,0hL(Ω)α1(1α1w1)Ch2|logh|2,

    which coincides with (4.3).

    (5.8) and (5.10) imply

    u1,1u1,1hL(Ω)u1,1w1,1hL(Ω)+u2,0u2,0hL(Ω) (5.13)

    with

    α1w1u1,1u1,1hL(Ω)+α1(1w1)u1,0u1,0hL(Ω)u2,0u2,0hL(Ω).

    Then, multiplying (5.13) by α1w1 and adding α1(1w1)u1,0u1,0hL(Ω), we obtain

    α1w1u1,1u1,1hL(Ω)+α1(1w1)u1,0u1,0hL(Ω)α1w1u1,1w1,1hL(Ω)+α1w1u2,0u2,0hL(Ω)+α1(1w1)u1,0u1,0hL(Ω).

    We note that

    α1w1u1,1u1,1hL(Ω)+α1(1w1)u1,0u1,0hL(Ω)

    is bounded by both

    α1w1u1,1w1,1hL(Ω)+α1w1u2,0u2,0hL(Ω)+α1(1w1)u1,0u1,0hL(Ω)

    and

    u2,0u2,0hL(Ω).

    So,

    α1w1u1,1w1,1hL(Ω)+α1w1u2,0u2,0hL(Ω)+α1(1w1)u1,0u1,0hL(Ω)u2,0u2,0hL(Ω)

    or

    u2,0u2,0hL(Ω)α1w1u1,1w1,1hL(Ω)+α1w1u2,0u2,0hL(Ω)+α1(1w1)u1,0u1,0hL(Ω).

    Therefore, according to (5.5) and (4.3), we get

    α1w1(1α1w1)u1,1w1,1hL(Ω)+α1(1w1)(1α1w1)u1,0u1,0hL(Ω)u2,0u2,0hL(Ω)Ch2|logh|2

    or

    u2,0u2,0hL(Ω)α1w1(1α1w1)u1,1w1,1hL(Ω)+α1(1w1)(1α1w1)u1,0u1,0hL(Ω)α1(1α1w1)Ch2|logh|2.

    So, the last two alternatives are true at the same time because both coincide with (4.3). We necessarily deduce that

    u2,0u2,0hL(Ω)=α1w1(1α1w1)u1,1w1,1hL(Ω)+α1(1w1)(1α1w1)u1,0u1,0hL(Ω). (5.14)

    By replacing (5.14) in (5.13); we get (5.12). Hence, in both cases (5.9) and (5.10); we can write

    u1,1u1,1hL(Ω)max1i2{1(1αiwi)}max1i2ui,1wi,1hL(Ω)
    +max1i2{αi(1wi)(1αiwi)}max1i2ui,0ui,0hL(Ω).

    Thus,

    u1,1u1,1hL(Ω)(γ+ρ)maxn0max1i2ui,nwi,nhL(Ω). (5.15)

    ● In a similar way, that is by following the same steps as for u1,1 and u1,1h, u2,1 and u2,1h satisfy

    u2,1u2,1hL(Ω)u2,1w2,1hL(Ω)+w2,1hu2,1hL(Ω).

    So,

    u2,1u2,1hL(Ω)u2,1w2,1hL(Ω)
    +max{α2w2u2,1u2,1hL(Ω)+α2(1w2)u2,0u2,0hL(Ω);u1,1u1,1hL(Ω)}. (5.16)

    We distinguish also two cases

    max{α2w2u2,1u2,1hL(Ω)+α2(1w2)u2,0u2,0hL(Ω);u1,1u1,1hL(Ω)}=α2w2u2,1u2,1hL(Ω)+α2(1w2)u2,0u2,0hL(Ω) (5.17)

    or

    max{α2w2u2,1u2,1hL(Ω)+α2(1w2)u2,0u2,0hL(Ω);u1,1u1,1hL(Ω)}=u1,1u1,1hL(Ω). (5.18)

    (5.16) in conjunction with case (5.17); we get

    u2,1u2,1hL(Ω)u2,1w2,1hL(Ω)+α2w2u2,1u2,1hL(Ω)

    +α2(1w2)u2,0u2,0hL(Ω)

    with

    u1,1u1,1hL(Ω)α2w2u2,1u2,1hL(Ω)+α2(1w2)u2,0u2,0hL(Ω). (5.19)

    So,

    u2,1u2,1hL(Ω)1(1α2w2)u2,1w2,1hL(Ω)+α2(1w2)(1α2w2)u2,0u2,0hL(Ω) (5.20)

    with, according to (5.20)

    u1,1u1,1hL(Ω)1(1α2w2)u2,1w2,1hL(Ω)+α2(1w2)(1α2w2)u2,0u2,0hL(Ω).

    Then,

    u1,1u1,1hL(Ω)max1i2{1(1αiwi)}ui,1wi,1hL(Ω)+max1i2{αi(1wi)(1αiwi)}ui,0ui,0hL(Ω).

    Therefore,

    u1,1u1,1hL(Ω)(γ+ρ)maxn0max1i2ui,nwi,nhL(Ω),

    which coincides with (5.15). The conjunction of (5.16) with case (5.18), implies

    u2,1u2,1hL(Ω)u2,1w2,1hL(Ω)+u1,1u1,1hL(Ω) (5.21)

    with

    α2w2u2,1u2,1hL(Ω)+α2(1w2)u2,0u2,0hL(Ω)u1,1u1,1hL(Ω).

    Then, by multiplying (5.21) by α2w2 and addingα2(1w2)u2,0u2,0hL(Ω), we obtain that the term α2w2u2,1u2,1hL(Ω)+α2(1w2)u2,0u2,0hL(Ω)is bounded by both

    α2w2u2,1w2,1hL(Ω)+α2w2u1,1u1,1hL(Ω)+α2(1w2)u2,0u2,0hL(Ω)

    and

    u1,1u1,1hL(Ω).

    So, we distinguish again, the two following alternatives

    α2w2(1α2w2)u2,1w2,1hL(Ω)+α2(1w2)(1α2w2)u2,0u2,0hL(Ω)u1,1u1,1hL(Ω)(γ+ρ)maxn0max1i2ui,nwi,nhL(Ω)

    or

    u1,1u1,1hL(Ω)α2w2(1α2w2)u2,1w2,1hL(Ω)+α2(1w2)(1α2w2)u2,0u2,0hL(Ω)(γ+ρ)maxn0max1i2ui,nwi,nhL(Ω).

    We remark that both alternatives coincide with (5.15), which implies that case (5.18) is possible if and only if

    u1,1u1,1hL(Ω)=α2w2(1α2w2)u2,1w2,1hL(Ω)+α2(1w2)(1α2w2)u2,0u2,0hL(Ω). (5.22)

    By substituting (5.22) in (5.21), we get (5.20). Hence, in both cases (5.17) and (5.18), we get

    u2,1u2,1hL(Ω)max1i2{1(1αiwi)}max1i2ui,1wi,1hL(Ω)+max1i2{αi(1wi)(1αiwi)}max1i2ui,0ui,0hL(Ω).

    Thus,

    u2,1u2,1hL(Ω)(γ+ρ)maxn0max1i2ui,nwi,nhL(Ω). (5.23)

    (5.15) and (5.23) imply

    U1U1h(γ+ρ)maxn0UnWnh.

    ● Let us assume that for n0 and i = 1, 2

    ui,nui,nhL(Ω)(γ(1+ρ++ρn1)+ρn)maxn0max1i2ui,nwi,nhL(Ω). (5.24)

    ● And prove for i = 1, 2

    ui,n+1ui,n+1hL(Ω)(γ(1+ρ++ρn)+ρn+1)maxn0max1i2ui,nwi,nhL(Ω). (5.25)

    We operate in the same way as in iterate n = 0. Let us begin with case i = 1 in (5.25)

    u1,n+1u1,n+1hL(Ω)u1,n+1w1,n+1hL(Ω)+w1,n+1hu1,n+1hL(Ω).

    So, by applying (2.17), we get

    u1,n+1u1,n+1hL(Ω)u1,n+1w1,n+1hL(Ω)
    +max{α1w1u1,n+1u1,n+1hL(Ω)+α1(1w1)u1,nu1,nhL(Ω);u2,nu2,nhL(Ω)} (5.26)

    We distinguish again two cases

    max{α1w1u1,n+1u1,n+1hL(Ω)+α1(1w1)u1,nu1,nhL(Ω);u2,nu2,nhL(Ω)}
    =α1w1u1,n+1u1,n+1hL(Ω)+α1(1w1)u1,nu1,nhL(Ω) (5.27)

    or

    max{α1w1u1,n+1u1,n+1hL(Ω)+α1(1w1)u1,nu1,nhL(Ω);u2,nu2,nhL(Ω)}
    =u2,nu2,nhL(Ω). (5.28)

    (5.26) in conjunction with case (5.27) implies

    u1,n+1u1,n+1hL(Ω)u1,n+1w1,n+1hL(Ω)+α1w1u1,n+1u1,n+1hL(Ω)+α1(1w1)u1,nu1,nhL(Ω)

    and

    u2,nu2,nhL(Ω)α1w1u1,n+1u1,n+1hL(Ω)+α1(1w1)u1,nu1,nhL(Ω).

    Then,

    u1,n+1u1,n+1hL(Ω)1(1α1w1)u1,n+1w1,n+1hL(Ω)+α1(1w1)(1α1w1)u1,nu1,nhL(Ω) (5.29)

    with, according to (5.29)

    u2,nu2,nhL(Ω)α1w1(1α1w1)u1,n+1w1,n+1hL(Ω)+α1(1w1)(1α1w1)u1,nu1,nhL(Ω).

    (5.24) implies

    u1,n+1u1,n+1hL(Ω)1(1α1w1)u1,n+1w1,n+1hL(Ω)+α1(1w1)(1α1w1)((γ(1+ρ++ρn1)+ρn)maxn0max1i2ui,nwi,nhL(Ω))

    with

    u2,nu2,nhL(Ω)α1w1(1α1w1)u1,n+1w1,n+1hL(Ω)
    +α1(1w1)(1α1w1)((γ(1+ρ++ρn1)+ρn)maxn0max1i2ui,nwi,nhL(Ω)).

    Thus,

    u1,n+1u1,n+1hL(Ω)γu1,n+1w1,n+1hL(Ω)+ρ((γ(1+ρ++ρn1)+ρn)maxn0max1i2ui,nwi,nhL(Ω))

    and as α1w1<1

    u2,nu2,nhL(Ω)
    γu1,n+1w1,n+1hL(Ω)+ρ((γ(1+ρ++ρn1)+ρn)maxn0max1i2ui,nwi,nhL(Ω)).

    Hence,

    u1,n+1u1,n+1hL(Ω)(γ(1+ρ++ρn)+ρn+1)maxn0max1i2ui,nwi,nhL(Ω)

    and

    u2,nu2,nhL(Ω)(γ(1+ρ++ρn)+ρn+1)maxn0max1i2ui,nwi,nhL(Ω).

    which corresponds with (5.24) for i = 2: Inequality (5.26) with (5.28) imply

    u1,n+1u1,n+1hL(Ω)u1,n+1w1,n+1hL(Ω)+u2,nu2,nhL(Ω) (5.30)

    and

    α1w1u1,n+1u1,n+1hL(Ω)+α1(1w1)u1,nu1,nhL(Ω)u2,nu2,nhL(Ω).

    By multiplying (5.30) by α1w1 and adding the term α1(1w1)u1,nu1,nhL(Ω), we get that the term

    α1w1u1,n+1u1,n+1hL(Ω)+α1(1w1)u1,nu1,nhL(Ω)

    is bounded by the following two terms

    α1w1u1,n+1w1,n+1hL(Ω)+α1w1u2,nu2,nhL(Ω)+α1(1w1)u1,nu1,nhL(Ω)

    and

    u2,nu2,nhL(Ω).

    So, we need to distinguish the followings possibilities

    α1w1u1,n+1w1,n+1hL(Ω)+α1w1u2,nu2,nhL(Ω)+α1(1w1)u1,nu1,nhL(Ω)u2,nu2,nhL(Ω)

    or

    u2,nu2,nhL(Ω)α1w1u1,n+1w1,n+1hL(Ω)+α1w1u2,nu2,nhL(Ω)+α1(1w1)u1,nu1,nhL(Ω),

    which implies

    α1w1(1α1w1)u1,n+1w1,n+1hL(Ω)+α1(1w1)(1α1w1)u1,nu1,nhL(Ω)u2,nu2,nhL(Ω)

    or

    u2,nu2,nhL(Ω)α1w1(1α1w1)u1,n+1w1,n+1hL(Ω)+α1(1w1)(1α1w1)u1,nu1,nhL(Ω).

    By using (5.24), we can write

    α1w1(1α1w1)u1,n+1w1,n+1hL(Ω)+α1(1w1)(1α1w1)u1,nu1,nhL(Ω)u2,nu2,nhL(Ω)(γ(1+ρ++ρn1)+ρn)maxn0max1i2ui,nwi,nhL(Ω),

    or

    u2,nu2,nhL(Ω)α1w1(1α1w1)u1,n+1w1,n+1hL(Ω)+α1(1w1)(1α1w1)u1,nu1,nhL(Ω)(γ(1+ρ++ρn1)+ρn)maxn0max1i2ui,nwi,nhL(Ω).

    Only the last alternative is true because it matches with (5.24) for i = 2. So, in (5.28) we get

    u2,nu2,nhL(Ω)α1w1(1α1w1)u1,n+1w1,n+1hL(Ω)+α1(1w1)(1α1w1)u1,nu1,nhL(Ω). (5.31)

    By replacing (5.31) in (5.30); we get (5.29). Hence, in both cases (5.27) and (5.28), we obtain

    u1,n+1u1,n+1hL(Ω)max1i2{1(1αiwi)}max1i2ui,n+1wi,n+1hL(Ω)+max1i2{αi(1wi)(1αiwi)}max1i2ui,nui,nhL(Ω).

    So,

    u1,n+1u1,n+1hL(Ω)γmaxn0max1i2ui,nwi,nhL(Ω)+ρ(γ(1+ρ++ρn1)+ρn)maxn0max1i2ui,nwi,nhL(Ω).

    Therefore,

    u1,n+1u1,n+1hL(Ω)(γ(1+ρ++ρn1+ρn)+ρn+1)maxn0max1i2ui,nwi,nhL(Ω). (5.32)

    By using the last inequality (5.32) and by adopting the same reasoning we prove (5.25); i = 2, therefore, we get (5.6).

    Theorem 2 Let U and Uh be the solution of systems (1.1) and (4.8), respectively. Then, there exists a constant C independent of h such that

    UUhγ1ρh2|logh|2.

    Proof. Making use of (3.6), (5.6) and (4.8), we have

    UUhUUn+1+Un+1Un+1h+Un+1hUh
    ρn+1UU0+(γ(1ρn+11ρ)+ρn+1)maxn0UnWnh+ρn+1UhU0h.

    As n+ and by using (5.5) we get (5.33).

    In this work an optimal convergence order is derived for a class of system of two elliptic quasi-variational inequalities where terms sources and obstacles depend upon the solution, where the continuous and discrete Lipschitz dependence with respect to the terms sources, boundary condition and obstacles' played a leading role in obtaining the main result of this paper. As (1.1) plays a key role in solving Hamilton-Jacobi-Bellman equation the results obtained in this paper can give an optimal error estimate for HJB equation also even for J2. The approach used and the results obtained in this paper (optimal convergence order) remain valid when we deal with systems of J2 quasi-variational inequalities with terms sources depends on solution and the obstacles i independent of the solution, that is systems of the form; Find a vector U=(u1,,uJ)(H10(Ω))J satisfying

    {ai(ui,vui)(fi(ui),vui);vH10(Ω)v,uiψi;ui0andi=1,,J.

    The author states that no funding source or sponsor has participated in the realization of this work.

    All authors declare no conflicts of interest in this paper.



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