Green intelligent building materials is an effective way for building materials industry to reduce carbon. However, a small amount of research and development (R&D, unstable R&D investment and imperfect collaborative innovation mode hinder the development of green intelligent building materials industry. However, few scholars study the development mechanism of green intelligent building materials industry from the perspective of industrial chain considering the above obstacles. In this study, the game models under market mechanism and government regulation were constructed to analyze the income distribution mechanism for the development mechanism of green intelligent building materials industry. Finally, the questionnaire method was used to discuss the game strategy of collaborative innovation behavior among agents. The results are as follows. In the game strategy selection of collaborative innovation behavior among green intelligent building materials, factors such as database marketing maturity, information flow and technology volume generated by collaborative innovation, technical benefit coefficient, social benefit coefficient and profit and loss barrier factors are conducive to the collaborative innovation behavior of green intelligent building materials. When the market mechanism fails, the incentive effect of cost subsidy adopted by the government is more efficient and fast, and the driving force of achievement reward is more lasting. The combination of the two incentives is the best. Moderate supervision and punishment lower than the free rider income can not ensure fair competition among green intelligent building materials enterprises. The punishment above the threshold can effectively restrain the negative impact of free rider income and prospect profit and loss. This study not only theoretically expands the development theory of digital industry from the perspective of industrial chain by considering the maturity factor of database, but also provides policy guidance for the development of green intelligent building materials industry in practice.
Citation: Chengli Hu, Ping Liu, Hongtao Yang, Shi Yin, Kifayat Ullah. A novel evolution model to investigate the collaborative innovation mechanism of green intelligent building materials enterprises for construction 5.0[J]. AIMS Mathematics, 2023, 8(4): 8117-8143. doi: 10.3934/math.2023410
[1] | Saudia Jabeen, Bandar Bin-Mohsin, Muhammad Aslam Noor, Khalida Inayat Noor . Inertial projection methods for solving general quasi-variational inequalities. AIMS Mathematics, 2021, 6(2): 1075-1086. doi: 10.3934/math.2021064 |
[2] | S. S. Chang, Salahuddin, M. Liu, X. R. Wang, J. F. Tang . Error bounds for generalized vector inverse quasi-variational inequality Problems with point to set mappings. AIMS Mathematics, 2021, 6(2): 1800-1815. doi: 10.3934/math.2021108 |
[3] | Shujie Jing, Jixiang Guan, Zhiyong Si . A modified characteristics projection finite element method for unsteady incompressible Magnetohydrodynamics equations. AIMS Mathematics, 2020, 5(4): 3922-3951. doi: 10.3934/math.2020254 |
[4] | Muhammad Aslam Noor, Khalida Inayat Noor, Bandar B. Mohsen . Some new classes of general quasi variational inequalities. AIMS Mathematics, 2021, 6(6): 6406-6421. doi: 10.3934/math.2021376 |
[5] | Jamilu Abubakar, Poom Kumam, Jitsupa Deepho . Multistep hybrid viscosity method for split monotone variational inclusion and fixed point problems in Hilbert spaces. AIMS Mathematics, 2020, 5(6): 5969-5992. doi: 10.3934/math.2020382 |
[6] | Zuliang Lu, Xiankui Wu, Fei Huang, Fei Cai, Chunjuan Hou, Yin Yang . Convergence and quasi-optimality based on an adaptive finite element method for the bilinear optimal control problem. AIMS Mathematics, 2021, 6(9): 9510-9535. doi: 10.3934/math.2021553 |
[7] | Chunjuan Hou, Zuliang Lu, Xuejiao Chen, Fei Huang . Error estimates of variational discretization for semilinear parabolic optimal control problems. AIMS Mathematics, 2021, 6(1): 772-793. doi: 10.3934/math.2021047 |
[8] | Safeera Batool, Muhammad Aslam Noor, Khalida Inayat Noor . Merit functions for absolute value variational inequalities. AIMS Mathematics, 2021, 6(11): 12133-12147. doi: 10.3934/math.2021704 |
[9] | Zuliang Lu, Fei Cai, Ruixiang Xu, Chunjuan Hou, Xiankui Wu, Yin Yang . A posteriori error estimates of hp spectral element method for parabolic optimal control problems. AIMS Mathematics, 2022, 7(4): 5220-5240. doi: 10.3934/math.2022291 |
[10] | Satit Saejung . A counterexample to the new iterative scheme of Rezapour et al.: Some discussions and corrections. AIMS Mathematics, 2023, 8(4): 9436-9442. doi: 10.3934/math.2023475 |
Green intelligent building materials is an effective way for building materials industry to reduce carbon. However, a small amount of research and development (R&D, unstable R&D investment and imperfect collaborative innovation mode hinder the development of green intelligent building materials industry. However, few scholars study the development mechanism of green intelligent building materials industry from the perspective of industrial chain considering the above obstacles. In this study, the game models under market mechanism and government regulation were constructed to analyze the income distribution mechanism for the development mechanism of green intelligent building materials industry. Finally, the questionnaire method was used to discuss the game strategy of collaborative innovation behavior among agents. The results are as follows. In the game strategy selection of collaborative innovation behavior among green intelligent building materials, factors such as database marketing maturity, information flow and technology volume generated by collaborative innovation, technical benefit coefficient, social benefit coefficient and profit and loss barrier factors are conducive to the collaborative innovation behavior of green intelligent building materials. When the market mechanism fails, the incentive effect of cost subsidy adopted by the government is more efficient and fast, and the driving force of achievement reward is more lasting. The combination of the two incentives is the best. Moderate supervision and punishment lower than the free rider income can not ensure fair competition among green intelligent building materials enterprises. The punishment above the threshold can effectively restrain the negative impact of free rider income and prospect profit and loss. This study not only theoretically expands the development theory of digital industry from the perspective of industrial chain by considering the maturity factor of database, but also provides policy guidance for the development of green intelligent building materials industry in practice.
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,v−ui)≥(fi(ui),v−ui);v∈H10(Ω) | (1.1) |
v,ui≤Mui;ui≥0. |
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+infμ≠iuμ; |
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),1≤i≤2 sufficiently smooth functions such that 1≤j,k≤N
∑1≤j,k≤Naijk(x)ξjξk≥α|ξ|2,ξ∈RN,α>0 |
ai0(x)≥βi>0,(x∈Ω) | (2.1) |
where βi is a positive constant. We define the bilinear forms: For all u,v∈H10(Ω)
ai(u,v)=∫Ω(∑1≤j,k≤Naijk(x)∂u∂xj∂v∂xk+∑Nk=1aik(x)∂u∂xkv+ai0(x)uv)dx | (2.2) |
We are given right-hand sides
fisuchthatfi∈L∞(Ω),fi≥f0>0, |
a nonlinear functional and Lipschitz continuous on R; that is
|fi(x)−fi(y)|≤ki|x−y|,∀x,y∈R, |
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‖∞=max1≤i≤2‖wi‖L∞(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
f∈L∞(Ω), | (2.5) |
the obstacle
ψ∈W2,∞(Ω)andψ≥0, | (2.6) |
the boundary condition g∈L∞(∂Ω) and the nonempty convex set
Kg={v∈H1(Ω)suchthatv=gon∂Ωandv≤ψonΩ}. | (2.7) |
We consider the variational inequality V.I.: Find u∈Kg such that
a(u,v−u)≥(f,v−u),∀v∈Kg. | (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),∀v≥0withζ_≤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≤˜gon∂Ω. |
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−˜ζ‖L∞(Ω)≤max{(1β)‖f−˜f‖L∞(Ω),‖g−˜g‖L∞(∂Ω),‖ψ−˜ψ‖L∞(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−˜f‖L∞(Ω),‖g−˜g‖L∞(∂Ω),‖ψ−˜ψ‖L∞(Ω)}. | (2.10) |
Then,
˜f≤f+‖f−˜f‖L∞(Ω) |
≤f+(1)‖f−˜f‖L∞(Ω) |
≤f+(a0(x)β)‖f−˜f‖L∞(Ω) |
≤f+a0(x)max{(1β)‖f−˜f‖L∞(Ω),‖g−˜g‖L∞(∂Ω),‖ψ−˜ψ‖L∞(Ω)}. |
So,
≤f+a0(x)φinΩ. | (2.11) |
Thus, for all 0<v,
(˜f,v)≤(f+a0(x)φ,v), |
with
˜ζ≤˜g≤g+φon∂Ω, |
˜ζ≤˜ψ≤ψ+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
˜ζ≤ζ+φin¯Ω |
or
˜ζ−ζ≤φin¯Ω. | (2.12) |
Similarly, interchanging the roles of the couples (f,g,ψ); (˜f,˜g,˜ψ), we obtain
ζ−˜ζ≤φin¯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 uh∈Kgh such that
a(uh,v−uh)≥(f,v−uh),∀v∈Kgh. | (2.14) |
Where
Kgh={v∈Vhsuchthatv=πhgon∂Ωandv≤rhψonΩ}, | (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
‖ζ−ζh‖L∞(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 rhψ≤rh˜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−˜ζh‖L∞(Ω)≤max{(1β)‖f−˜f‖L∞(Ω),‖g−˜g‖L∞(∂Ω),‖rhψ−rh˜ψ‖L∞(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 ζi∈H10(Ω)∩L∞(Ω) is a solution to the following variational inequality
ai(ζi,v−ζi)≥(fi(zi),v−ζi);v∈H10(Ω) | (3.1) |
v,ζi≤Mζi=k+zj;ζi≥0withi≠j. |
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),∀v∈H10(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,v−u1,n+1)≥(w1f1(u1,n+1)+(1−w1)f1(u1,n),v−u1,n+1) | (3.2) |
v,u1,n+1≤Mu1,n+1=k+u2,n, | (3.3) |
a2(u2,n+1,v−u2,n+1)≥(w2f2(u2,n+1)+(1−w2)f2(u2,n),v−u2,n+1) | (3.4) |
v,u2,n+1≤Mu2,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 n≥0
‖Un+1−U‖∞≤ρn+1‖U0−U‖∞ | (3.6) |
where
ρ=max1≤i≤2α1(1−w1)1−α1w1<1. | (3.7) |
Proof. The proof will carry out by induction.
● We first deal with the case
‖u1−u1,0‖L∞(Ω)=max1≤i≤2‖ui−ui,0‖L∞(Ω). | (3.8) |
● Indeed for n = 0; using (1.1), (3.2), (3.3) and (2.9), we have
‖u1−u1,1‖L∞(Ω)≤max{(1β1)‖f1(u1)−(w1f1(u1,1)+(1−w1)f1(u1,0))‖L∞(Ω);‖u2−u2,0‖L∞(Ω)} |
≤max{(1β1)‖w1(f1(u1)−f1(u1,1))+(1−w1)(f1(u1)−f1(u1,0))‖L∞(Ω);‖u2−u2,0‖L∞(Ω)} |
≤max{(k1β1)(w1‖u1−u1,1‖L∞(Ω)+(1−w1)‖u1−u1,0‖L∞(Ω));‖u2−u2,0‖L∞(Ω)}. |
So,
‖u1−u1,1‖L∞(Ω)≤max{α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω);‖u2−u2,0‖L∞(Ω)} | (3.9) |
We distinguish two cases
max{α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω);‖u2−u2,0‖L∞(Ω)} |
=α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω) | (3.10) |
or
max{α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω);‖u2−u2,0‖L∞(Ω)} |
=‖u2−u2,0‖L∞(Ω) | (3.11) |
(3.9) in conjunction with case (3.10) implies
‖u1−u1,1‖L∞(Ω)≤α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω) | (3.12) |
with
‖u2−u2,0‖L∞(Ω)≤α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω), | (3.13) |
which implies
‖u1−u1,1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω). | (3.14) |
By replacing (3.14) in (3.13), we get
‖u2−u2,0‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω) |
≤ρmax1≤i≤2‖ui−ui,0‖L∞(Ω), |
which coincides with (3.8).
(3.9) in conjunction with (3.11) implies
‖u1−u1,1‖L∞(Ω)≤‖u2−u2,0‖L∞(Ω) | (3.15) |
with
α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω)≤‖u2−u2,0‖L∞(Ω). | (3.16) |
‖u2−u2,0‖L∞(Ω) is bounded below by both ‖u1−u1,1‖L∞(Ω)
and
α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω). |
So,
‖u1−u1,1‖L∞(Ω)≤α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω) |
or
α1w1‖u1−u1,1‖L∞(Ω)+α1(1−w1)‖u1−u1,0‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω). |
Then,
‖u1−u1,1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω) | (3.17) |
or
α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω). | (3.18) |
(3.15), (3.17) and (3.18) generate the following three possibilities
‖u1−u1,1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω)≤‖u2−u2,0‖L∞(Ω)≤max1≤i≤2‖ui−ui,0‖L∞(Ω) |
or
‖u1−u1,1‖L∞(Ω)≤‖u2−u2,0‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω)≤max1≤i≤2‖ui−ui,0‖L∞(Ω) |
or
α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω)≤‖u2−u2,0‖L∞(Ω)≤max1≤i≤2‖ui−ui,0‖L∞(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
‖u1−u1,1‖L∞(Ω)=α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω). |
Hence, both cases (3.10) and (3.11) imply (3.14).
● Let us now discuss the second case
‖u2−u2,0‖L∞(Ω)=max1≤i≤2‖ui−ui,0‖L∞(Ω). | (3.19) |
(3.9) in conjunction with (3.10) implies (3.14) with
‖u2−u2,0‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω) |
≤ρmax1≤i≤2‖ui−ui,0‖L∞(Ω)<‖u2−u2,0‖L∞(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,
‖u1−u1,1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω)≤max1≤i≤2‖ui−ui,0‖L∞(Ω) |
or
α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω)≤max1≤i≤2‖ui−ui,0‖L∞(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
‖u1−u1,1‖L∞(Ω)=α1(1−w1)1−α1w1‖u1−u1,0‖L∞(Ω). |
Hence, in both cases (3.8) and (3.19), we obtain (3.14). Hence,
‖u1−u1,1‖L∞(Ω)≤ρmax1≤i≤2‖ui−ui,0‖L∞(Ω). | (3.20) |
● As
U1=(u1,1,u2,1)andU=(u1,u2), |
we need to deal also with ‖u2−u2,1‖L∞(Ω), by following the same reasoning as that adopted for u1 and u1,1, we get
‖u2−u2,1‖L∞(Ω)≤max{α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω);‖u1−u1,1‖L∞(Ω)} | (3.21) |
Again we distinguish two possibilities
max{α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω);‖u1−u1,1‖L∞(Ω)} |
=α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω); | (3.22) |
or
max{α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω);‖u1−u1,1‖L∞(Ω)} |
=‖u1−u1,1‖L∞(Ω). | (3.23) |
(3.21) and (3.22) imply
‖u2−u2,1‖L∞(Ω)≤α2(1−w2)(1−α2w2)‖u2−u2,0‖L∞(Ω) | (3.24) |
with
‖u1−u1,1‖L∞(Ω)≤α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω). | (3.25) |
By substituting (3.24) in (3.25), we get
‖u1−u1,1‖L∞(Ω)≤α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω)≤ρmax1≤i≤2‖ui−ui,0‖L∞(Ω), |
which coincides with (3.20). (3.21) and (3.23) imply
‖u2−u2,1‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω), | (3.26) |
with
α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω). |
It is clear that ‖u1−u1,1‖L∞(Ω) is bounded below by both
‖u2−u2,1‖L∞(Ω) |
and
α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω), |
which leads us to distinguish the following possibilities
‖u2−u2,1‖L∞(Ω)≤α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω) |
or
α2w2‖u2−u2,1‖L∞(Ω)+α2(1−w2)‖u2−u2,0‖L∞(Ω)≤‖u2−u2,1‖L∞(Ω). |
Then,
‖u2−u2,1‖L∞(Ω)≤α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω) | (3.27) |
or
α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω)≤‖u2−u2,1‖L∞(Ω). | (3.28) |
Thus, (3.26)-(3.28) imply that the three following alternatives are required
‖u2−u2,1‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω)≤α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω) |
or
‖u2−u2,1‖L∞(Ω)≤α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω) |
or
α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω)≤‖u2−u2,1‖L∞(Ω)≤‖u1−u1,1‖L∞(Ω). |
It is clear that all alternatives coincide with (3.20). So, we must have
‖u2−u2,1‖L∞(Ω)=α2(1−w2)1−α2w2‖u2−u2,0‖L∞(Ω). |
Thus, in both cases (3.22) and (3.23) we obtain (3.24). Hence,
‖u2−u2,1‖L∞(Ω)≤ρmax1≤i≤2‖ui−ui,0‖L∞(Ω). | (3.29) |
(3.20) and (3.29) imply
‖U1−U‖∞≤ρ‖U0−U‖∞. |
● Let us assume that, for n≥0
‖ui−ui,n‖L∞(Ω)≤ρnmax1≤i≤2‖ui−ui,0‖L∞(Ω),i=1,2. | (3.30) |
● We prove
‖ui−ui,n+1‖L∞(Ω)≤ρn+1max1≤i≤2‖ui−ui,n‖L∞(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
‖u1−u1,n+1‖L∞(Ω)≤max{(1β1)‖f1(u1)−(w1f1(u1,n+1)+(1−w1)f1(u1,n))‖L∞(Ω);‖u2−u2,n‖L∞(Ω)} |
So,
‖u1−u1,n+1‖L∞(Ω)≤max{α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω);‖u2−u2,n‖L∞(Ω)} | (3.32) |
Also we distinguish two cases:
max{α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω);‖u2−u2,n‖L∞(Ω)} |
=α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω) | (3.33) |
or
max{α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω);‖u2−u2,n‖L∞(Ω)}=‖u2−u2,n‖L∞(Ω) | (3.34) |
(3.32) in conjunction with (3.33) implies
‖u1−u1,n+1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω), | (3.35) |
with
‖u2−u2,n‖L∞(Ω)≤α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω). | (3.36) |
By replacing (3.35) in (3.36) we get, according to (3.30); i = 1
‖u2−u2,n‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω)≤ρn+1max1≤i≤2‖ui−ui,0‖L∞(Ω) |
which matches with (3.30); i = 2. (3.32) in conjunction with (3.34) implies
‖u1−u1,n+1‖L∞(Ω)≤‖u2−u2,n‖L∞(Ω) | (3.37) |
with
α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω)≤‖u2−u2,n‖L∞(Ω). |
‖u2−u2,n‖L∞(Ω) is bounded below by both ‖u1−u1,n+1‖L∞(Ω)
and
α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω) |
So,
‖u1−u1,n+1‖L∞(Ω)≤α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω) |
or
α1w1‖u1−u1,n+1‖L∞(Ω)+α1(1−w1)‖u1−u1,n‖L∞(Ω)≤‖u1−u1,n+1‖L∞(Ω). |
Thus,
‖u1−u1,n+1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω) |
or
α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω)≤‖u1−u1,n+1‖L∞(Ω). |
By taking into account (3.37), we get
‖u1−u1,n+1‖L∞(Ω)≤‖u2−u2,n‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω) |
or
‖u1−u1,n+1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω)≤‖u2−u2,n‖L∞(Ω) |
or
α1(1−w1)1−α1w1‖u1−u1,n‖L∞(Ω)≤‖u1−u1,n+1‖L∞(Ω)≤‖u2−u2,n‖L∞(Ω). |
Three possibilities are true because all coincide with (3.30). So, we necessarily get
‖u1−u1,n+1‖L∞(Ω)≤α1(1−w1)1−α1w1‖u1−u1,n‖L∞(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,v−uih)≥(fi(uih),v−uih);v∈Vhv,uih≤rh(Muih)=rh(k+ujh);i≠j.uih≥0anduih=πhgon∂Ω. | (4.1) |
The related discrete fixed-point mapping
Th:(Vh)2→(Vh)2 |
Zh=(z1h,z2h)→ThZh=ζh=(ζ1h,ζ2h), |
where ζih∈Vh is the unique solution to the following discrete variational inequality
ai(ζih,v−ζih)≥(fi(zih),v−ζih);v∈Vh | (4.2) |
v,ζih≤rh(Mζih)=rh(k+zjh);ζih≥0withi≠jandζih=πhgon∂Ω. |
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,0−ui,0h‖L∞(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,v−u1,n+1h)≥(w1f1(u1,n+1h)+(1−w1)f1(u1,nh),v−u1,n+1h) | (4.4) |
v,u1,n+1h≤rh(Mu1,n+1h)=rh(k+u2,nh), | (4.5) |
a2(u2,n+1h,v−u2,n+1h)≥(w2f2(u2,n+1h)+(1−w2)f2(u2,nh),v−u2,n+1h) | (4.6) |
v,u2,n+1h≤rh(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 n≥0
‖Un+1h−Uh‖∞≤ρn+1‖U0h−Uh‖∞. | (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,v−w1,n+1h)≥(w1f1(u1,n+1)+(1−w1)f1(u1,n),v−w1,n+1h) | (5.1) |
v,w1,n+1h≤rh(Mu1,n+1)=rh(k+u2,n), | (5.2) |
a2(w2,n+1h,v−w2,n+1h)≥(w2f2(u2,n+1)+(1−w2)f2(u2,n),v−w2,n+1h) | (5.3) |
v,w2,n+1h≤rh(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+1h−ui,n+1‖L∞(Ω)≤Ch2|log|2;i=1,2andn≥0. | (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+1−Un+1h‖∞≤(γ(1−ρn+11−ρ)+ρn+1)maxn≥0‖Un−Wnh‖∞. | (5.6) |
Where
γ=max1≤i≤2{1(1−αiwi)}. | (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
‖u1,1−u1,1h‖L∞(Ω)≤‖u1,1−w1,1h‖L∞(Ω)+‖w1,1h−u1,1h‖L∞(Ω). |
(5.1), (5.2), (4.4), (4.5) and (2.17) imply
‖u1,1−u1,1h‖L∞(Ω)≤‖u1,1−w1,1h‖L∞(Ω) |
+max{(1β1)‖f1(u1,1)−(w1f1(u1,1h)+(1−w1)f1(u1,0h))‖L∞(Ω);‖rh(k+u2,0)−rh(k+u2,0h)‖L∞(Ω)} |
So,
‖u1,1−u1,1h‖L∞(Ω)≤‖u1,1−w1,1h‖L∞(Ω) |
+max{(k1β1)w1‖u1,1−u1,1h‖L∞(Ω)+(k1β1)(1−w1)‖u1,0−u1,0h‖L∞(Ω);‖rh(k+u2,0)−rh(k+u2,0h)‖L∞(Ω)}. |
Therefore,
‖u1,1−u1,1h‖L∞(Ω)≤‖u1,1−w1,1h‖L∞(Ω) | (5.8) |
+max{α1w1‖u1,1−u1,1h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω);‖u2,0−u2,0h‖L∞(Ω)}. |
We distinguish two cases
max{α1w1‖u1,1−u1,1h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω);‖u2,0−u2,0h‖L∞(Ω)} |
=α1w1‖u1,1−u1,1h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω) | (5.9) |
or
max{α1w1‖u1,1−u1,1h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω);‖u2,0−u2,0h‖L∞(Ω)}=‖u2,0−u2,0h‖L∞(Ω) | (5.10) |
(5.8) in conjunction with (5.9) imply
‖u1,1−u1,1h‖L∞(Ω)≤‖u1,1−w1,1h‖L∞(Ω)+α1w1‖u1,1−u1,1h‖L∞(Ω) |
+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω) |
with
‖u2,0−u2,0h‖L∞(Ω)≤α1w1‖u1,1−u1,1h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω). | (5.11) |
So,
(1−α1w1)‖u1,1−u1,1h‖L∞(Ω)≤‖u1,1−w1,1h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω), |
with (5.11). Then,
‖u1,1−u1,1h‖L∞(Ω)≤1(1−α1w1)‖u1,1−w1,1h‖L∞(Ω)+α1(1−w1)(1−α1w1)‖u1,0−u1,0h‖L∞(Ω). | (5.12) |
By replacing (5.12) in (5.11) we obtain
‖u2,0−u2,0h‖L∞(Ω)≤α1w1(1−α1w1)‖u1,1−w1,1h‖L∞(Ω)+α1(1−w1)(1−α1w1)‖u1,0−u1,0h‖L∞(Ω). |
According to (5.5) and (4.3) we get,
‖u2,0−u2,0h‖L∞(Ω)≤α1(1−α1w1)Ch2|logh|2, |
which coincides with (4.3).
(5.8) and (5.10) imply
‖u1,1−u1,1h‖L∞(Ω)≤‖u1,1−w1,1h‖L∞(Ω)+‖u2,0−u2,0h‖L∞(Ω) | (5.13) |
with
α1w1‖u1,1−u1,1h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω)≤‖u2,0−u2,0h‖L∞(Ω). |
Then, multiplying (5.13) by α1w1 and adding α1(1−w1)‖u1,0−u1,0h‖L∞(Ω), we obtain
α1w1‖u1,1−u1,1h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω)≤α1w1‖u1,1−w1,1h‖L∞(Ω)+α1w1‖u2,0−u2,0h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω). |
We note that
α1w1‖u1,1−u1,1h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω) |
is bounded by both
α1w1‖u1,1−w1,1h‖L∞(Ω)+α1w1‖u2,0−u2,0h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω) |
and
‖u2,0−u2,0h‖L∞(Ω). |
So,
α1w1‖u1,1−w1,1h‖L∞(Ω)+α1w1‖u2,0−u2,0h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω)≤‖u2,0−u2,0h‖L∞(Ω) |
or
‖u2,0−u2,0h‖L∞(Ω)≤α1w1‖u1,1−w1,1h‖L∞(Ω)+α1w1‖u2,0−u2,0h‖L∞(Ω)+α1(1−w1)‖u1,0−u1,0h‖L∞(Ω). |
Therefore, according to (5.5) and (4.3), we get
α1w1(1−α1w1)‖u1,1−w1,1h‖L∞(Ω)+α1(1−w1)(1−α1w1)‖u1,0−u1,0h‖L∞(Ω)≤‖u2,0−u2,0h‖L∞(Ω)≤Ch2|logh|2 |
or
‖u2,0−u2,0h‖L∞(Ω)≤α1w1(1−α1w1)‖u1,1−w1,1h‖L∞(Ω)+α1(1−w1)(1−α1w1)‖u1,0−u1,0h‖L∞(Ω)≤α1(1−α1w1)Ch2|logh|2. |
So, the last two alternatives are true at the same time because both coincide with (4.3). We necessarily deduce that
‖u2,0−u2,0h‖L∞(Ω)=α1w1(1−α1w1)‖u1,1−w1,1h‖L∞(Ω)+α1(1−w1)(1−α1w1)‖u1,0−u1,0h‖L∞(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
‖u1,1−u1,1h‖L∞(Ω)≤max1≤i≤2{1(1−αiwi)}max1≤i≤2‖ui,1−wi,1h‖L∞(Ω) |
+max1≤i≤2{αi(1−wi)(1−αiwi)}max1≤i≤2‖ui,0−ui,0h‖L∞(Ω). |
Thus,
‖u1,1−u1,1h‖L∞(Ω)≤(γ+ρ)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(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
‖u2,1−u2,1h‖L∞(Ω)≤‖u2,1−w2,1h‖L∞(Ω)+‖w2,1h−u2,1h‖L∞(Ω). |
So,
‖u2,1−u2,1h‖L∞(Ω)≤‖u2,1−w2,1h‖L∞(Ω) |
+max{α2w2‖u2,1−u2,1h‖L∞(Ω)+α2(1−w2)‖u2,0−u2,0h‖L∞(Ω);‖u1,1−u1,1h‖L∞(Ω)}. | (5.16) |
We distinguish also two cases
max{α2w2‖u2,1−u2,1h‖L∞(Ω)+α2(1−w2)‖u2,0−u2,0h‖L∞(Ω);‖u1,1−u1,1h‖L∞(Ω)}=α2w2‖u2,1−u2,1h‖L∞(Ω)+α2(1−w2)‖u2,0−u2,0h‖L∞(Ω) | (5.17) |
or
max{α2w2‖u2,1−u2,1h‖L∞(Ω)+α2(1−w2)‖u2,0−u2,0h‖L∞(Ω);‖u1,1−u1,1h‖L∞(Ω)}=‖u1,1−u1,1h‖L∞(Ω). | (5.18) |
(5.16) in conjunction with case (5.17); we get
‖u2,1−u2,1h‖L∞(Ω)≤‖u2,1−w2,1h‖L∞(Ω)+α2w2‖u2,1−u2,1h‖L∞(Ω)
+α2(1−w2)‖u2,0−u2,0h‖L∞(Ω) |
with
‖u1,1−u1,1h‖L∞(Ω)≤α2w2‖u2,1−u2,1h‖L∞(Ω)+α2(1−w2)‖u2,0−u2,0h‖L∞(Ω). | (5.19) |
So,
‖u2,1−u2,1h‖L∞(Ω)≤1(1−α2w2)‖u2,1−w2,1h‖L∞(Ω)+α2(1−w2)(1−α2w2)‖u2,0−u2,0h‖L∞(Ω) | (5.20) |
with, according to (5.20)
‖u1,1−u1,1h‖L∞(Ω)≤1(1−α2w2)‖u2,1−w2,1h‖L∞(Ω)+α2(1−w2)(1−α2w2)‖u2,0−u2,0h‖L∞(Ω). |
Then,
‖u1,1−u1,1h‖L∞(Ω)≤max1≤i≤2{1(1−αiwi)}‖ui,1−wi,1h‖L∞(Ω)+max1≤i≤2{αi(1−wi)(1−αiwi)}‖ui,0−ui,0h‖L∞(Ω). |
Therefore,
‖u1,1−u1,1h‖L∞(Ω)≤(γ+ρ)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω), |
which coincides with (5.15). The conjunction of (5.16) with case (5.18), implies
‖u2,1−u2,1h‖L∞(Ω)≤‖u2,1−w2,1h‖L∞(Ω)+‖u1,1−u1,1h‖L∞(Ω) | (5.21) |
with
α2w2‖u2,1−u2,1h‖L∞(Ω)+α2(1−w2)‖u2,0−u2,0h‖L∞(Ω)≤‖u1,1−u1,1h‖L∞(Ω). |
Then, by multiplying (5.21) by α2w2 and addingα2(1−w2)‖u2,0−u2,0h‖L∞(Ω), we obtain that the term α2w2‖u2,1−u2,1h‖L∞(Ω)+α2(1−w2)‖u2,0−u2,0h‖L∞(Ω)is bounded by both
α2w2‖u2,1−w2,1h‖L∞(Ω)+α2w2‖u1,1−u1,1h‖L∞(Ω)+α2(1−w2)‖u2,0−u2,0h‖L∞(Ω) |
and
‖u1,1−u1,1h‖L∞(Ω). |
So, we distinguish again, the two following alternatives
α2w2(1−α2w2)‖u2,1−w2,1h‖L∞(Ω)+α2(1−w2)(1−α2w2)‖u2,0−u2,0h‖L∞(Ω)≤‖u1,1−u1,1h‖L∞(Ω)≤(γ+ρ)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω) |
or
‖u1,1−u1,1h‖L∞(Ω)≤α2w2(1−α2w2)‖u2,1−w2,1h‖L∞(Ω)+α2(1−w2)(1−α2w2)‖u2,0−u2,0h‖L∞(Ω)≤(γ+ρ)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω). |
We remark that both alternatives coincide with (5.15), which implies that case (5.18) is possible if and only if
‖u1,1−u1,1h‖L∞(Ω)=α2w2(1−α2w2)‖u2,1−w2,1h‖L∞(Ω)+α2(1−w2)(1−α2w2)‖u2,0−u2,0h‖L∞(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
‖u2,1−u2,1h‖L∞(Ω)≤max1≤i≤2{1(1−αiwi)}max1≤i≤2‖ui,1−wi,1h‖L∞(Ω)+max1≤i≤2{αi(1−wi)(1−αiwi)}max1≤i≤2‖ui,0−ui,0h‖L∞(Ω). |
Thus,
‖u2,1−u2,1h‖L∞(Ω)≤(γ+ρ)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω). | (5.23) |
(5.15) and (5.23) imply
‖U1−U1h‖∞≤(γ+ρ)maxn≥0‖Un−Wnh‖∞. |
● Let us assume that for n≥0 and i = 1, 2
‖ui,n−ui,nh‖L∞(Ω)≤(γ(1+ρ+⋯+ρn−1)+ρn)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω). | (5.24) |
● And prove for i = 1, 2
‖ui,n+1−ui,n+1h‖L∞(Ω)≤(γ(1+ρ+⋯+ρn)+ρn+1)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(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+1−u1,n+1h‖L∞(Ω)≤‖u1,n+1−w1,n+1h‖L∞(Ω)+‖w1,n+1h−u1,n+1h‖L∞(Ω). |
So, by applying (2.17), we get
‖u1,n+1−u1,n+1h‖L∞(Ω)≤‖u1,n+1−w1,n+1h‖L∞(Ω) |
+max{α1w1‖u1,n+1−u1,n+1h‖L∞(Ω)+α1(1−w1)‖u1,n−u1,nh‖L∞(Ω);‖u2,n−u2,nh‖L∞(Ω)} | (5.26) |
We distinguish again two cases
max{α1w1‖u1,n+1−u1,n+1h‖L∞(Ω)+α1(1−w1)‖u1,n−u1,nh‖L∞(Ω);‖u2,n−u2,nh‖L∞(Ω)} |
=α1w1‖u1,n+1−u1,n+1h‖L∞(Ω)+α1(1−w1)‖u1,n−u1,nh‖L∞(Ω) | (5.27) |
or
max{α1w1‖u1,n+1−u1,n+1h‖L∞(Ω)+α1(1−w1)‖u1,n−u1,nh‖L∞(Ω);‖u2,n−u2,nh‖L∞(Ω)} |
=‖u2,n−u2,nh‖L∞(Ω). | (5.28) |
(5.26) in conjunction with case (5.27) implies
‖u1,n+1−u1,n+1h‖L∞(Ω)≤‖u1,n+1−w1,n+1h‖L∞(Ω)+α1w1‖u1,n+1−u1,n+1h‖L∞(Ω)+α1(1−w1)‖u1,n−u1,nh‖L∞(Ω) |
and
‖u2,n−u2,nh‖L∞(Ω)≤α1w1‖u1,n+1−u1,n+1h‖L∞(Ω)+α1(1−w1)‖u1,n−u1,nh‖L∞(Ω). |
Then,
‖u1,n+1−u1,n+1h‖L∞(Ω)≤1(1−α1w1)‖u1,n+1−w1,n+1h‖L∞(Ω)+α1(1−w1)(1−α1w1)‖u1,n−u1,nh‖L∞(Ω) | (5.29) |
with, according to (5.29)
‖u2,n−u2,nh‖L∞(Ω)≤α1w1(1−α1w1)‖u1,n+1−w1,n+1h‖L∞(Ω)+α1(1−w1)(1−α1w1)‖u1,n−u1,nh‖L∞(Ω). |
(5.24) implies
‖u1,n+1−u1,n+1h‖L∞(Ω)≤1(1−α1w1)‖u1,n+1−w1,n+1h‖L∞(Ω)+α1(1−w1)(1−α1w1)((γ(1+ρ+⋯+ρn−1)+ρn)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω)) |
with
‖u2,n−u2,nh‖L∞(Ω)≤α1w1(1−α1w1)‖u1,n+1−w1,n+1h‖L∞(Ω) |
+α1(1−w1)(1−α1w1)((γ(1+ρ+⋯+ρn−1)+ρn)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω)). |
Thus,
‖u1,n+1−u1,n+1h‖L∞(Ω)≤γ‖u1,n+1−w1,n+1h‖L∞(Ω)+ρ((γ(1+ρ+⋯+ρn−1)+ρn)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω)) |
and as α1w1<1
‖u2,n−u2,nh‖L∞(Ω) |
≤γ‖u1,n+1−w1,n+1h‖L∞(Ω)+ρ((γ(1+ρ+⋯+ρn−1)+ρn)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω)). |
Hence,
‖u1,n+1−u1,n+1h‖L∞(Ω)≤(γ(1+ρ+⋯+ρn)+ρn+1)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω) |
and
‖u2,n−u2,nh‖L∞(Ω)≤(γ(1+ρ+⋯+ρn)+ρn+1)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω). |
which corresponds with (5.24) for i = 2: Inequality (5.26) with (5.28) imply
‖u1,n+1−u1,n+1h‖L∞(Ω)≤‖u1,n+1−w1,n+1h‖L∞(Ω)+‖u2,n−u2,nh‖L∞(Ω) | (5.30) |
and
α1w1‖u1,n+1−u1,n+1h‖L∞(Ω)+α1(1−w1)‖u1,n−u1,nh‖L∞(Ω)≤‖u2,n−u2,nh‖L∞(Ω). |
By multiplying (5.30) by α1w1 and adding the term α1(1−w1)‖u1,n−u1,nh‖L∞(Ω), we get that the term
α1w1‖u1,n+1−u1,n+1h‖L∞(Ω)+α1(1−w1)‖u1,n−u1,nh‖L∞(Ω) |
is bounded by the following two terms
α1w1‖u1,n+1−w1,n+1h‖L∞(Ω)+α1w1‖u2,n−u2,nh‖L∞(Ω)+α1(1−w1)‖u1,n−u1,nh‖L∞(Ω) |
and
‖u2,n−u2,nh‖L∞(Ω). |
So, we need to distinguish the followings possibilities
α1w1‖u1,n+1−w1,n+1h‖L∞(Ω)+α1w1‖u2,n−u2,nh‖L∞(Ω)+α1(1−w1)‖u1,n−u1,nh‖L∞(Ω)≤‖u2,n−u2,nh‖L∞(Ω) |
or
‖u2,n−u2,nh‖L∞(Ω)≤α1w1‖u1,n+1−w1,n+1h‖L∞(Ω)+α1w1‖u2,n−u2,nh‖L∞(Ω)+α1(1−w1)‖u1,n−u1,nh‖L∞(Ω), |
which implies
α1w1(1−α1w1)‖u1,n+1−w1,n+1h‖L∞(Ω)+α1(1−w1)(1−α1w1)‖u1,n−u1,nh‖L∞(Ω)≤‖u2,n−u2,nh‖L∞(Ω) |
or
‖u2,n−u2,nh‖L∞(Ω)≤α1w1(1−α1w1)‖u1,n+1−w1,n+1h‖L∞(Ω)+α1(1−w1)(1−α1w1)‖u1,n−u1,nh‖L∞(Ω). |
By using (5.24), we can write
α1w1(1−α1w1)‖u1,n+1−w1,n+1h‖L∞(Ω)+α1(1−w1)(1−α1w1)‖u1,n−u1,nh‖L∞(Ω)≤‖u2,n−u2,nh‖L∞(Ω)≤(γ(1+ρ+⋯+ρn−1)+ρn)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω), |
or
‖u2,n−u2,nh‖L∞(Ω)≤α1w1(1−α1w1)‖u1,n+1−w1,n+1h‖L∞(Ω)+α1(1−w1)(1−α1w1)‖u1,n−u1,nh‖L∞(Ω)≤(γ(1+ρ+⋯+ρn−1)+ρn)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω). |
Only the last alternative is true because it matches with (5.24) for i = 2. So, in (5.28) we get
‖u2,n−u2,nh‖L∞(Ω)≤α1w1(1−α1w1)‖u1,n+1−w1,n+1h‖L∞(Ω)+α1(1−w1)(1−α1w1)‖u1,n−u1,nh‖L∞(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+1−u1,n+1h‖L∞(Ω)≤max1≤i≤2{1(1−αiwi)}max1≤i≤2‖ui,n+1−wi,n+1h‖L∞(Ω)+max1≤i≤2{αi(1−wi)(1−αiwi)}max1≤i≤2‖ui,n−ui,nh‖L∞(Ω). |
So,
‖u1,n+1−u1,n+1h‖L∞(Ω)≤γmaxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω)+ρ(γ(1+ρ+⋯+ρn−1)+ρn)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(Ω). |
Therefore,
‖u1,n+1−u1,n+1h‖L∞(Ω)≤(γ(1+ρ+⋯+ρn−1+ρn)+ρn+1)maxn≥0max1≤i≤2‖ui,n−wi,nh‖L∞(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
‖U−Uh‖∞≤γ1−ρh2|logh|2. |
Proof. Making use of (3.6), (5.6) and (4.8), we have
‖U−Uh‖∞≤‖U−Un+1‖∞+‖Un+1−Un+1h‖∞+‖Un+1h−Uh‖∞ |
≤ρn+1‖U−U0‖∞+(γ(1−ρn+11−ρ)+ρn+1)maxn≥0‖Un−Wnh‖∞+ρn+1‖Uh−U0h‖∞. |
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 J≥2. The approach used and the results obtained in this paper (optimal convergence order) remain valid when we deal with systems of J≥2 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,v−ui)≥(fi(ui),v−ui);v∈H10(Ω)v,ui≤ψi;ui≥0andi=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.
[1] |
C. Debrah, A. P. C. Chan, A. Darko, Artificial intelligence in green building, Autom. Constr., 137 (2022), 104192. https://doi.org/10.1016/j.autcon.2022.104192 doi: 10.1016/j.autcon.2022.104192
![]() |
[2] |
X. Q. Liu, C. A. Wang, X. M. Zhang, L. Gao, J. N. Zhu, Financing constraints change of China's green industries, AIMS Mathematics, 7 (2022), 20873–20890. https://doi.org/10.3934/math.20221144 doi: 10.3934/math.20221144
![]() |
[3] |
Y. Su, H. Y. Cheng, Z. Wang, L. W. Wang, Impacts of the COVID-19 lockdown on building energy consumption and indoor environment: A case study in Dalian, China, Energ. Buildings, 263 (2022), 112055. https://doi.org/10.1016/j.enbuild.2022.112055 doi: 10.1016/j.enbuild.2022.112055
![]() |
[4] |
X. C. Zeng, S. C. Li, S. Yin, Z. Y. Xing, How does the government promote the collaborative innovation of green building projects? An evolutionary game perspective, Buildings, 12 (2022), 1179. https://doi.org/10.3390/buildings12081179 doi: 10.3390/buildings12081179
![]() |
[5] |
P. F. Pereira, N. M. Ramos, M. L. Simões, Data-driven occupant actions prediction to achieve an intelligent building, Build. Res. Inf., 48 (2020), 485–500. https://doi.org/10.1080/09613218.2019.1692648 doi: 10.1080/09613218.2019.1692648
![]() |
[6] |
C. X. Jia, H. Y. Ding, C. J. Zhang, X. Zhang, Design of a dynamic key management plan for intelligent building energy management system based on wireless sensor network and blockchain technology, Alex. Eng. J., 60 (2021), 337–346. https://doi.org/10.1016/j.aej.2020.08.019 doi: 10.1016/j.aej.2020.08.019
![]() |
[7] |
T. D. Shao, Indoor environment intelligent control system of green building based on PMV index, Adv. Civ. Eng., 2021 (2021), 6619401. https://doi.org/10.1155/2021/6619401 doi: 10.1155/2021/6619401
![]() |
[8] |
S. Yin, B. Z. Li, Z. Y. Xing, The governance mechanism of the building material industry (BMI) in transformation to green BMI: The perspective of green building, Sci. Total Environ., 677 (2019), 19–33. https://doi.org/10.1016/j.scitotenv.2019.04.317 doi: 10.1016/j.scitotenv.2019.04.317
![]() |
[9] |
J. Lu, K. Wang, M. L. Qu, Experimental determination on the capillary water absorption coefficient of porous building materials: A comparison between the intermittent and continuous absorption tests, J. Build. Eng., 28 (2020), 101091. https://doi.org/10.1016/j.jobe.2019.101091 doi: 10.1016/j.jobe.2019.101091
![]() |
[10] |
S. Yin, N. Zhang, H. M. Dong, Preventing COVID-19 from the perspective of industrial information integration: Evaluation and continuous improvement of information networks for sustainable epidemic prevention, J. Ind. Inf. Integr., 19 (2020), 100157. https://doi.org/10.1016/j.jii.2020.100157 doi: 10.1016/j.jii.2020.100157
![]() |
[11] |
S. Yin, N. Zhang, K. Ullah, S. Gao, Enhancing digital innovation for the sustainable transformation of manufacturing industry: A pressure-state-response system framework to perceptions of digital green innovation and its performance for green and intelligent manufacturing, Systems, 10 (2022), 72. https://doi.org/10.3390/systems10030072 doi: 10.3390/systems10030072
![]() |
[12] |
M. A. Omer, T. Noguchi, A conceptual framework for understanding the contribution of building materials in the achievement of Sustainable Development Goals (SDGs), Sustain. Cities Soc., 52 (2020), 101869. https://doi.org/10.1016/j.scs.2019.101869 doi: 10.1016/j.scs.2019.101869
![]() |
[13] |
T. Dong, S. Yin, N. Zhang, New energy-driven construction industry: Digital green innovation investment project selection of photovoltaic building materials enterprises using an integrated fuzzy decision approach, Syst., 11 (2023), 11. https://doi.org/10.3390/systems11010011 doi: 10.3390/systems11010011
![]() |
[14] |
F. J. Kong, L. H. He, Impacts of supply-sided and demand-sided policies on innovation in green building technologies: A case study of China, J. Clean. Prod., 294 (2021), 126279. https://doi.org/10.1016/j.jclepro.2021.126279 doi: 10.1016/j.jclepro.2021.126279
![]() |
[15] |
A. K. Kan, N. Zheng, W. B. Zhu, D. Cao, W. Wang, Innovation and development of vacuum insulation panels in China: A state-of-the-art review, J. Build. Eng., 48 (2021), 103937. https://doi.org/10.1016/j.jobe.2021.103937 doi: 10.1016/j.jobe.2021.103937
![]() |
[16] |
W. A. Medeiros, M. de Oliveira Soriani, G. A. Parsekian, Innovation in flat-jack application to evaluate modern high-strength hollow concrete block masonry, Constr. Build. Mater., 255 (2020), 119341. https://doi.org/10.1016/j.conbuildmat.2020.119341 doi: 10.1016/j.conbuildmat.2020.119341
![]() |
[17] |
Y. F. Jiang, W. Y. Zheng, Coupling mechanism of green building industry innovation ecosystem based on blockchain smart city, J. Clean. Prod., 307 (2021), 126766. https://doi.org/10.1016/j.jclepro.2021.126766 doi: 10.1016/j.jclepro.2021.126766
![]() |
[18] |
R. D. Lumpkin, T. W. Horton, V. J. Sinfield, Holistic synergy analysis for building subsystem performance and innovation opportunities, Build. Environ., 178 (2020), 106908. https://doi.org/10.1016/j.buildenv.2020.106908 doi: 10.1016/j.buildenv.2020.106908
![]() |
[19] |
B. Lv, X. G. Qi, Research on partner combination selection of the supply chain collaborative product innovation based on product innovative resources, Comput. Ind. Eng., 128 (2019), 245–253. https://doi.org/10.1016/j.cie.2018.12.041 doi: 10.1016/j.cie.2018.12.041
![]() |
[20] | H. Y. Guo, C. J. Fan, J. C. Li, L. Wang, H. C. Wu, Y. P. Yang, Study on co-evolution of e-commerce industry and big data industry considering internal and external factors, Oper. Res. Manag., 28 (2019), 191–199. http://www.jorms.net/EN/10.12005/orms.2019.0072 |
[21] |
R. K. P. Maddikunta, V. Q. Pham, B. Prabadevi, N. Deepa, K. Dev, R. T. Gadekallu, et al., Industry 5.0: A survey on enabling technologies and potential applications, J. Ind. Inf. Integr., 26 (2021), 100257. https://doi.org/10.1016/j.jii.2021.100257 doi: 10.1016/j.jii.2021.100257
![]() |
[22] |
A. Mojumder, A. Singh, An exploratory study of the adaptation of green supply chain management in construction industry: the case of Indian Construction Companies, J. Clean. Prod., 295 (2021), 126400. https://doi.org/10.1016/j.jclepro.2021.126400 doi: 10.1016/j.jclepro.2021.126400
![]() |
[23] |
M. M. W. Wijesiri, K. A. K. Devapriya, P. Rathnasiri, L. T. Wickremanayake Karunaratne, A framework to implement green adaptive reuse for existing buildings in Sri Lanka, Intell. Build. Int., 14 (2021), 581–605. https://doi.org/10.1080/17508975.2021.1906204 doi: 10.1080/17508975.2021.1906204
![]() |
[24] |
G. Wang, Y. Li, J. Zuo, W. B. Hu, Q. W. Nie, H. Q. Lei, Who drives green innovations? Characteristics and policy implications for green building collaborative innovation networks in China, Renew. Sust. Energ. Rev., 143 (2021), 110875. https://doi.org/10.1016/j.rser.2021.110875 doi: 10.1016/j.rser.2021.110875
![]() |
[25] |
X. W. Li, R. N. Huang, J. C. Dai, J. R. Li, Q. Shen, Research on the evolutionary game of construction and demolition waste (CDW) recycling units' green behavior, considering remanufacturing capability, Int. J. Environ. Res. Public. Health., 18 (2021), 9268. https://doi.org/10.3390/ijerph18179268 doi: 10.3390/ijerph18179268
![]() |
[26] |
M. Jeihoonian, M. Kazemi Zanjani, M. Gendreau, Dynamic reverse supply chain network design under uncertainty: Mathematical modeling and solution algorithm, Int. T. Oper. Res., 29 (2022), 3161–3189. https://doi.org/10.1111/itor.12865 doi: 10.1111/itor.12865
![]() |
[27] |
S. Yin, B. Z. Li, Academic research institutes-construction enterprises linkages for the development of urban green building: Selecting management of green building technologies innovation partner, Sustain. Cities Soc., 48 (2019), 101555. https://doi.org/10.1016/j.scs.2019.101555 doi: 10.1016/j.scs.2019.101555
![]() |
[28] |
L. H. He, L. Y. Chen, The incentive effects of different government subsidy policies on green buildings, Renew. Sust. Energ. Rev., 135 (2021), 110123. https://doi.org/10.1016/j.rser.2020.110123 doi: 10.1016/j.rser.2020.110123
![]() |
[29] | A. Karamikli, Y. Bayar, Impact of information and communication technology on CO2 emissions: Evidence from EU transition economies, In: Technological development and impact on economic and environmental sustainability, New York: IGI Global Press, 2022. |
[30] |
S. Yin, N. Zhang, J. F. Xu, Information fusion for future COVID-19 prevention: Continuous mechanism of big data intelligent innovation for the emergency management of a public epidemic outbreak, J. Manag. Anal., 8 (2021), 391–423. https://doi.org/10.1080/23270012.2021.1945499 doi: 10.1080/23270012.2021.1945499
![]() |
[31] |
S. S. Guo, B. G. Du, L. B. Sun, Y. B. Li, J. Guo, Design and implementation of digital management platform for building materials equipment manufacturing enterprises, CIMS, 21 (2015), 226–234. https://doi.org/10.13196/j.cims.2015.01.025 doi: 10.13196/j.cims.2015.01.025
![]() |
[32] | M. S. Zhang, Y. Cui, Investigation and research on Shenzhen green building materials market, New Build. Mater., 45 (2018), 143–146. |
[33] |
L. L. He, H. Yuan, Research on quality perception of recycled building materials enterprises, Ind. Eng. Manag., 24 (2019), 144–151. https://doi.org/10.19495/j.cnki.1007-5429.2019.01.019 doi: 10.19495/j.cnki.1007-5429.2019.01.019
![]() |
[34] |
W. Wang, Z. Tian, W. Xi, Y. R. Tan, Y. Deng, The influencing factors of China's green building development: An analysis using RBF-WINGS method, Build. Environ., 188 (2021), 107425. https://doi.org/10.1016/j.buildenv.2020.107425 doi: 10.1016/j.buildenv.2020.107425
![]() |
[35] |
S. Yin, T. Dong, B. Z. Li, S. Gao, Developing a conceptual partner selection framework: Digital green innovation management of prefabricated construction enterprises for sustainable urban development, Buildings, 12 (2022), 721. https://doi.org/10.3390/buildings12060721 doi: 10.3390/buildings12060721
![]() |
[36] |
Q. Liu, F. H. Gong, Research on e-commerce logistics service upgrading of traditional building materials professional market, Logist. Technol., 2014 (2014), 22–25. https://doi.org/10.3969/j.issn.1005-152X.2014.05.007 doi: 10.3969/j.issn.1005-152X.2014.05.007
![]() |
[37] |
C. Cohen, D. Pearlmutter, M. Schwartz, Promoting green building in Israel: A game theory-based analysis, Build. Environ., 163 (2019), 106227. https://doi.org/10.1016/j.buildenv.2019.106227 doi: 10.1016/j.buildenv.2019.106227
![]() |
[38] |
Y. Liu, J. Zuo, M. Pan, Q. Ge, R. D. Chang, X. T. Feng, et al., The incentive mechanism and decision-making behavior in the green building supply market: A tripartite evolutionary game analysis, Build. Environ., 214 (2022), 108903. https://doi.org/10.1016/j.buildenv.2022.108903 doi: 10.1016/j.buildenv.2022.108903
![]() |
[39] | H. Lintsen, Stagnation and dynamism in three supply chains: agriculture and foods, building materials and construction, energy, In: Well-being, sustainability and social development, Cham: Springer, 2018. https://doi.org/10.1007/978-3-319-76696-6 |
[40] |
M. Li, M. Lu, H. L. Yu, Research on building materials Information Technology Framework under block chain technology, Build. Econ., 40 (2019), 103–107. https://doi.org/10.14181/j.cnki.1002-851x.201910103 doi: 10.14181/j.cnki.1002-851x.201910103
![]() |
[41] |
M. A. Wibowo, N. U. Handayani, A. Mustikasari, S. A. Wardani, B. Tjahjono, Reverse logistics performance indicators for the construction sector: A building project case, Sustainability, 14 (2022), 963. https://doi.org/10.3390/su14020963 doi: 10.3390/su14020963
![]() |
[42] |
S. M. Khoshnava, R. Rostami, A. Valipour, M. Ismail, A. R. Rahmat, Rank of green building material criteria based on the three pillars of sustainability using the hybrid multi criteria decision making method, J. Clean. Prod., 173 (2018), 82–99. https://doi.org/10.1016/j.jclepro.2016.10.066 doi: 10.1016/j.jclepro.2016.10.066
![]() |
[43] |
A. Darko, A. P. Chan, X. S. Huo, D. G. Owusu-Manu, A scientometric analysis and visualization of global green building research, Build. Environ., 149 (2019), 501–511. https://doi.org/10.1016/j.buildenv.2018.12.059 doi: 10.1016/j.buildenv.2018.12.059
![]() |
[44] |
C. C. Menassa, From BIM to digital twins: A systematic review of the evolution of intelligent building representations in the AEC-FM industry, J. Inf. Technol. Constr., 26 (2021), 58–83. https://doi.org/10.36680/j.itcon.2021.005 doi: 10.36680/j.itcon.2021.005
![]() |
[45] |
Z. H. Mohson, Z. A. Ismael, S. S. Shalal, Comparison between smart and traditional building materials to achieve sustainability, Period. Eng. Nat. Sci., 9 (2021), 808–822. http://doi.org/10.21533/pen.v9i3.2283 doi: 10.21533/pen.v9i3.2283
![]() |
[46] |
S. Yin, Y. Y. Yu, An adoption-implementation framework of digital green knowledge to improve the performance of digital green innovation practices for industry 5.0, J. Clean. Prod., 363 (2022), 132608. https://doi.org/10.1016/j.jclepro.2022.132608 doi: 10.1016/j.jclepro.2022.132608
![]() |
[47] |
M. M. Wang, S. Lian, S. Yin, H. M. Dong, A three-player game model for promoting the diffusion of green technology in manufacturing enterprises from the perspective of supply and demand, Mathematics, 8 (2020), 1585. https://doi.org/10.3390/math8091585 doi: 10.3390/math8091585
![]() |