Loading [MathJax]/extensions/TeX/boldsymbol.js
Research article Special Issues

Identifying the volatility spillover risks between crude oil prices and China's clean energy market

  • Received: 28 August 2022 Revised: 08 October 2022 Accepted: 09 October 2022 Published: 21 October 2022
  • Since the COVID-19 outbreak, the global economy has been hit hard, and the development of renewable energy and energy transitions has become a common choice for all countries. The development of clean energy firms has become a hot topic of discussion among scholars, and the relationship between the stock prices of clean energy firms and the international crude oil market has attracted more attention. In this paper, we analyze the volatility connectedness between crude oil and Chinese clean energy firms from 2016 to 2022 by building time-varying vector autoregressive models with stochastic volatility components and time-varying spillover index and dynamic conditional correlation GARCH models. The results of the shock effects analysis show that international crude oil volatility had a significant short-term positive impact on Chinese clean energy firms during the COVID-19 outbreak period. Regarding spillover analysis, firms with large total market capitalization tended to be the senders of volatility spillovers, while smaller firms were likely to be the recipients. In terms of dynamic correlation analysis, the correlation between international crude oil and each clean energy firm was found to be volatile, and the dynamic correlation coefficient tended to reach its highest point during the COVID-19 outbreak. Meanwhile, from the optimal portfolio weighting analysis, it is clear that all optimal weights of international crude oil and medium clean energy firms will increase during an epidemic outbreak, and that more assets should be invested in clean energy firms.

    Citation: Hao Nong, Yitan Guan, Yuanying Jiang. Identifying the volatility spillover risks between crude oil prices and China's clean energy market[J]. Electronic Research Archive, 2022, 30(12): 4593-4618. doi: 10.3934/era.2022233

    Related Papers:

    [1] Shan Jiang, Li Liang, Meiling Sun, Fang Su . Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation. Electronic Research Archive, 2020, 28(2): 935-949. doi: 10.3934/era.2020049
    [2] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao . A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28(4): 1439-1457. doi: 10.3934/era.2020076
    [3] Jun Pan, Yuelong Tang . Two-grid H1-Galerkin mixed finite elements combined with L1 scheme for nonlinear time fractional parabolic equations. Electronic Research Archive, 2023, 31(12): 7207-7223. doi: 10.3934/era.2023365
    [4] Hongze Zhu, Chenguang Zhou, Nana Sun . A weak Galerkin method for nonlinear stochastic parabolic partial differential equations with additive noise. Electronic Research Archive, 2022, 30(6): 2321-2334. doi: 10.3934/era.2022118
    [5] Xiu Ye, Shangyou Zhang, Peng Zhu . A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29(1): 1897-1923. doi: 10.3934/era.2020097
    [6] Leilei Wei, Xiaojing Wei, Bo Tang . Numerical analysis of variable-order fractional KdV-Burgers-Kuramoto equation. Electronic Research Archive, 2022, 30(4): 1263-1281. doi: 10.3934/era.2022066
    [7] Derrick Jones, Xu Zhang . A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces. Electronic Research Archive, 2021, 29(5): 3171-3191. doi: 10.3934/era.2021032
    [8] Chunmei Wang . Simplified weak Galerkin finite element methods for biharmonic equations on non-convex polytopal meshes. Electronic Research Archive, 2025, 33(3): 1523-1540. doi: 10.3934/era.2025072
    [9] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang . A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, 2021, 29(3): 2375-2389. doi: 10.3934/era.2020120
    [10] Yan Yang, Xiu Ye, Shangyou Zhang . A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids. Electronic Research Archive, 2024, 32(5): 3413-3432. doi: 10.3934/era.2024158
  • Since the COVID-19 outbreak, the global economy has been hit hard, and the development of renewable energy and energy transitions has become a common choice for all countries. The development of clean energy firms has become a hot topic of discussion among scholars, and the relationship between the stock prices of clean energy firms and the international crude oil market has attracted more attention. In this paper, we analyze the volatility connectedness between crude oil and Chinese clean energy firms from 2016 to 2022 by building time-varying vector autoregressive models with stochastic volatility components and time-varying spillover index and dynamic conditional correlation GARCH models. The results of the shock effects analysis show that international crude oil volatility had a significant short-term positive impact on Chinese clean energy firms during the COVID-19 outbreak period. Regarding spillover analysis, firms with large total market capitalization tended to be the senders of volatility spillovers, while smaller firms were likely to be the recipients. In terms of dynamic correlation analysis, the correlation between international crude oil and each clean energy firm was found to be volatile, and the dynamic correlation coefficient tended to reach its highest point during the COVID-19 outbreak. Meanwhile, from the optimal portfolio weighting analysis, it is clear that all optimal weights of international crude oil and medium clean energy firms will increase during an epidemic outbreak, and that more assets should be invested in clean energy firms.



    We will present a weak Galerkin finite element method for the following parabolic singularly perturbed convection-reaction-diffusion problem:

    {tuεΔu+bu+cu=f(x,y,t)inΩ×(0,T],u=0onΩ×(0,T],u(x,0)=u0in¯Ω, (1.1)

    where Ω=(0,1)2, 0<ε1, and T>0 is a constant. Assume b=b(x,y),c=c(x,y), and u0=u0(x,y) are sufficiently smooth functions on Ω, and

    b=(b1(x,y),b2(x,y))(β1,β2),c12bc0>0on Ω, (1.2)

    for some constants β1,β2, and c0. The parabolic convection-dominated problem (1.1) has been utilized in a broad range of applied mathematics and engineering including fluid dynamics, electrical engineering, and the transport problem [1,2].

    In general, the solution of the problem (1.1) will have abrupt changes along the boundary. In other words, the solution exhibits boundary/interior layers near the boundary of Ω. We are only interested in the boundary layers by excluding the interior layers which can be accomplished by assuming some extra compatible conditions on the data; see, e.g., [1,3]. The standard numerical schemes including the finite element method give unsatisfactory numerical results due to the boundary layers. Some nonphysical oscillations in the numerical solution can occur even on adapted meshes, and it is not easy to solve efficiently the resulting discrete system [4]. There are many numerical schemes for solving convection-dominated problems efficiently and accurately in the literature. These methods include Galerkin finite element methods [5,6,7], weak Galerkin finite element methods (WG-FEMs) [8,9,10], the streamline upwind Petrov-Galerkin (SUPG) [11,12], and the discontinuous Galerkin (DG) methods [13,14,15]. Among these numerical methods, the standard WG-FEM introduced in [16] is also an effective and flexible numerical algorithm for solving PDEs. Recently, the WG methods demonstrate robust and stable discretizations for singularly perturbed problems. In fact, while the WG-FEM and the hybridizable discontinuous Galerkin share something in common, the WG-FEM seems more appropriate for solving the time dependent singularly perturbed problems since the inclusion of the convective term in the context of hybridized methods is not straightforward and makes the analysis more subtle. Errors estimates of arbitrary-order methods, including the virtual element method (VEM), are typically limited by the regularity of the exact solution. A distinctive feature of the WG-FEM lies in its use of weak function spaces. Moreover, hybrid high-order (HHO) methods have similar features with WG-FEMs. In fact, the reconstruction operator in the HHO method and the weak gradient operator in WG methods are closely related, and that the main difference between HHO and WG methods lies in the choice of the discrete unknowns and the design of the stabilization operator [17]. Notably, in weak Galerkin methods, weak derivatives are used instead of strong derivatives in variational form for underlying PDEs and adding parameter free stabilization terms. Considering the application of the WG method, various PDEs arising from the mathematical modeling of practical problems in science are solved numerically via WG-FEMs using the concept of weak derivatives. There exist many papers on such PDEs including elliptic equations in [16,18,19], parabolic equations [20,21,22], hyperbolic equations [23,24], etc.

    However, to the best of the author's knowledge, there is no work regarding the uniform convergence results of the fully-discrete WG-FEM for singularly perturbed parabolic problems on layer-adapted meshes. This paper uses three layer-adapted meshes defined through mesh generating functions, namely, Shishkin-type meshes, Bakhvalov-Shishkin type meshes and Bakhvalov-type meshes given in [25]. The error estimates in this work show that one has optimal order of convergence for Bakhvalov-Shishkin type meshes and Bakhvalov-type meshes while almost optimal convergence for Shishkin-type meshes. The main ingredient of the error analysis is the use of the vertices-edges interpolation of Lin [26]. The main advantage of this interpolation operator is that we have sharper error bounds compared with the classical interpolation operators. For the sake of simplicity, the Crank–Nicolson method is used for time discretization. This scheme yields optimal order estimates for fully-discrete WG-FEM. As an alternative, one can apply a discontinuous Galerkin method in time and present optimal order estimates for the fully-discrete scheme [27].

    The rest of the paper is organized as follows. In Section 2, we introduce some notations and recall some definitions. The regularity of the solution is also summarized and three layer-adapted meshes have been introduced in Section 2. Also, we define the weak gradient and weak convection operators, and using these concepts we define our bilinear forms. In Section 3, the semi-discrete WG-FEM and its stability results have been presented. Section 4 introduces a special interpolation operator and analyses interpolation error estimates. Section 5 presents error analysis of the semi-discrete WG-FEM for the problem (1.1) on the layer-adapted meshes. In Section 6, we apply the Crank-Nicolson scheme on uniform time mesh in time to obtain the fully-discrete WG-FEM, and prove uniform error estimates on the layer-adapted meshes. In Section 7, we conduct some numerical examples to validate the robustness of the WG-FEM for the problem (1.1). Summary on the contributions of this work are presented in Section 8.

    Let S be a measurable subset of Ω. We shall use the classical Sobolev spaces Wr,q(S),Hr(S)=Wr,2(S),Hr0(S),Lq(S)=W0,q(S) for negative integers r>0 and 1q, and (,)S for the L2 inner product on S. The semi-norm and norm on Hr(S) are denoted by ||r,S and r,S, respectively. If S=Ω, we do not write S in the subscript. Throughout the study, we shall use C as a positive generic constant, which is independent of the mesh parameters h and ε.

    This section deals with the introduction of a decomposition of the solution which provides a priori information on the exact solution and its derivatives. Based on this solution decomposition, we construct layer-adapted meshes. As we noted in the introduction, the solution of (1.1) exhibits typically two exponential boundary layers at x=1 and y=1. The following lemma gives some information on the solution decomposition and bounds on the solution of (1.1) and its derivatives.

    Lemma 2.1. Let k be positive integer and l(0,1). Assume that the solution u of the problem (1.1) belongs to the space Ck+l(QT) where QT:=Ω×(0,T]. Assume further that the solution u can be decomposed into a smooth part uR and layer components uL0,uL1, and uL1 with

    u=uR+uL,uL=uL0+uL1+uL2,(x,y)¯Ω, (2.1)

    where the smooth and layer parts satisfy

    |i+j+ruRixjytr(x,y)|C (2.2)
    |i+j+ruL0ixjytr(x,y)|Cεieβ1(1x)/ε, (2.3)
    |i+j+ruL1ixjytr(x,y)|Cεjeβ2(1y)/ε, (2.4)
    |i+j+ruL2ixjytr(x,y)|Cε(i+j)eβ1(1x)/εeβ2(1y)/ε, (2.5)

    for any (x,y)¯Ω,t[0,T], and positive integers i,j,r with i+j+2rk, and C only depends on b,c, and f and is independent of ε. Here, uR is the regular part of u, uL0 is an exponential boundary layer along the side x=1 of Ω, uL1 is an exponential boundary layer along the side y=1, while uL2 is an exponential corner layer at (1,1).

    Proof. Under some smoothness conditions and strong imposed compatibility requirements, Shishkin proved this solution decomposition; see, [1].

    Let Nx and Ny be positive integers. For the sake of simplicity, we assume that Nx=Ny=N is an even integer number. We shall construct the tensor product mesh TN={Tij}i,j=1,2,N in ¯Ω consisting of elements Tij=Ii×Kj with the intervals Ii=(xi1,xi) and Kj=(yj1,yj), where the mesh points are defined by

    0=x0<x1<,xNx=1,0=y0<y1<,yNy=1.

    Since the construction of the meshes in both directions is similar, the mesh construction in x-variable is given here.

    We define the transition parameter as

    τ=min(12,σεβ1φ(1/2)),

    where σp+1 is a positive constant. Here, p is the degree of the polynomials used in the approximation space. The function φ obeys the conditions

    φ(0)=0,φ>0,φ0. (2.6)

    Assumption 1. Throughout this article, we assume that εCN1 such that τ=(p+1)εβ1φ(1/2), since otherwise the analysis can be carried out using uniform mesh.

    Let the mesh points xi be equally distributed in [0,xN/2] with N/2 intervals and distributed [xN/2,1] with N/2 intervals using the mesh generating function defined by

    xi=λ(i/N)={2(1τ)i/N,for i=0,1,,N/2,1(k+1)εβ1φ(1i/N),for i=N/2,N/2+1,,N. (2.7)

    For example, as in [25], the Shishkin-type (S-type) meshes can be deduced by φ(1/2)=lnN while Bakhvalov-type meshes (B-type) can be recovered by taking φ(1/2)=ln(1/ε).

    We will use the mesh characterizing function ψ defined by ψ=eφ, which is an essential tool in our analysis below.

    Following [2], we list some famous adaptive meshes including S-type, Bakhvalov-Shishkin meshes (BS-mesh), and B-type in Table 1.

    Table 1.  Frequently used layer-adapted meshes.
    S-type BS B-type
    φ(t) 2tlnN ln[12(1N1)t] ln[12(1ε)t]
    ψ(t) N2t 12(1N1)t 12(1ε)t
    max|ψ| ClnN C C

     | Show Table
    DownLoad: CSV

    Similarly, we define the transition point in the y-direction as

    τy:=min(12,σεβ2φ(1/2)).

    We first split the domain Ω into four subdomains as in Figure 1:

    Ωr:=[0,1τ]×[0,1τy],Ωx:=[1τ,1]×[0,1τy],Ωy:=[0,1τ]×[1τy,1],Ωxy:=[1τ,1]×[1τy,1].
    Figure 1.  Tensor product Shishkin mesh for N=8.

    Clearly, the mesh is uniform in Ωr with a mesh size of O(N1), highly anisotropic in Ωx and Ωy, while it is very fine in Ωxy.

    Let hxi:=xixi1,i=1,,Nhyj:=yjyj1,j=1,,N be the mesh sizes of the subintervals. For the sake of simplicity, we assume that β1=β2=β. Then, one has hxi=hyj, and we simply write hi,i=1,,N for simplicity. The following technical lemmas show the smallness of the boundary layer-functions and the basic properties of the mesh sizes of the layer-adapted meshes.

    Lemma 2.2. [28] Denote by Θi=min{hi/ε,1}eα(1xi)/σε for i=N/2+1,,N. There exists a constant C>0 independent of ε and N such that

    maxN/2+1iNΘiCN1max|ψ|
    Ni=N/2+1ΘiC

    Lemma 2.3. [28] For the layer-adapted meshes we considered here, we have

    h1=h2==hN/2andmini=1,,NhiCεN1max|ψ|.

    Moreover, for i=N/2+1,,N,

    hi=2τ/N,for S-type1hi+1hiC,for BS- mesh

    and, for the B-type mesh,

    i=N/2+2,,N,1hi+1hiC,i=1,2,,N/2,hN/2+iσεβ1i+1,

    where C>0 is a constant independent of ε and N.

    A weak formulation of the problems (1.1) and (1.2) is to look for uH10(Ω) such that

    (ut,v)+A(u,v)=(f,v)vH10(Ω), (2.8)

    where the bilinear from A(,) is defined by

    A(u,v):=ε(u,v)+(bu,v)+(cu,v).

    Based on the weak formulation (2.8), we define the WG-FEM on the layer-adapted mesh. Let p be a positive integer. We define a local WG-FE space V(p,K) on each KTN given by

    V(p,K):={vN={v0,vb}:v0|KQp(K),vb|ePp(e),eK},

    where Qp(K) is the polynomials of degree p on K in both variables, and Pp(e) denotes the polynomials of degree p on the edge e.

    Defining the WG finite element space VN globally on TN as

    VN={vN={v0,vb}:vN|KV(p,K),vb|eK1=vb|eK2,K1K2={e}}, (2.9)

    and its subspace

    V0N={v:vVN,vb=0 on Ω}.

    The weak gradient operator wuN[Qp1(K)]2 can be defined on K as

    (wuN,ψ)K=(u0,ψ)K+ub,ψnKψ[Qp1(K)]2, (2.10)

    where n represents the outward unit normal K, (w,v)K denotes the inner product on K for functions w and v, and w,vK is the L2inner product on K.

    The weak convection operator bwuNQp(K) can be defined on K as

    (bwuN,ξ)K=(u0,(bξ))K+ub,bnξKξQp(K). (2.11)

    For simplicity, we adapt

    (ϕ,ψ)=KTN(ϕ,ψ)K,ϕ2=(ϕ,ϕ),ϕ,ψ=KTNϕ,ψK.

    For uN={u0,ub}VN and vN={v0,vb}VN, the bilinear form Aw(,) is given by

    Aw(uN,vN)=ε(wuN,wvN)+(bwuN,v0)+(cu0,vo)+Sd(uN,vN)+Sc(uN,vN), (2.12)

    where sd(,) and sc(,) are bilinear forms defined by

    Sd(uN,vN)=KTNρKu0ub,v0vbK,Sc(uN,vN)=KTNbn(u0ub),v0vbK+

    with K+={zK:b(z)n(z)0} and ρK is the penalty term given by

    ρK:={1,if KΩr,N(max|ψ|)1,if KΩΩr. (2.13)

    Given a mesh rectangle K, its dimensions parallel to the x and y-axes are written as hx,K and hy,K, respectively.

    Lemma 2.4. [29] For all KTN with hK=min{hx,K,hy,K}, there exists a constant C depending only on p such that

    uN2L2(K)Ch1KuN2L2(K),uNPp(K). (2.14)

    We next formulate our semi-discrete WG scheme as follows (Algorithm 1).

    Algorithm 1 The semi-discrete WG-FEM for problem (1.1)
    Find uN=(u0,ub)V0N such that
             (u0,v0)+Aw(uN,vN)=(f,v0)vN=(v0,vb)V0N,uN(0)=u0(0),     (2.15)
    where u0(0)V0N is an approximation of u(0).

    This section is devoted to establishing the stability results of the WG-FEM defined by (2.15). Define the energy norm E on the weak function space VN for vN={v0,vb}VN,

    vN2E:=εKTNwvN2K+KTN|bn|1/2(v0vb)2K+v02+Sd(vN,vN). (3.1)

    Define also an H1 equivalent norm on VN by

    vN2ε:=εKTNv02K+KTN|bn|1/2(v0vb)2K+v02+Sd(vN,vN). (3.2)

    The equivalence of these two norms on V0N is given in the next lemma.

    Lemma 3.1. For vNV0N, one has

    CvNεvNECvNε.

    Proof. For vN={v0,vb}V0N, it follows from the weak gradient operator (2.10) and integration by parts that;

    (wvN,w)K=(v0,w)Kv0vb,nwK,w[Qp1(K)]2,KTN. (3.3)

    Choosing w=wvN in (3.3) and using the Cauchy-Schwartz inequality and the trace inequality (4), we arrive at

    (wvN,wvN)K=(v0,wvN)Kv0vb,nwvNKv0L2(K)wvNL2(K)+v0vbL2(KwvNL2(K)(v0L2(K)+Ch1/2Kv0vbL2(K))wvNL2(K).

    Hence, we get

    wvNL2(K)v0L2(K)+Ch1/2Kv0vbL2(K).

    Summing over all KTN yields

    εwvN22(εv02+CKTNεh1Kv0vb2L2(K)).

    From the penalty term (2.13), we get

    εh1KρKC,KTN,

    and

    KTNεh1Kv0vbL2(K)=KTNεh1KρKρKv0vbL2(K)Csd(vN,vN).

    As a result, for vNV0N, we have

    vNECvNε. (3.4)

    Taking w=v0 in (3.3) and using the Cauchy-Schwartz inequality, we get

    (v0,v0)K=(wvN,v0)K+v0vb,nv0KwvNL2(K)v0L2(K)+v0vbL2(Kv0L2(K)(wvNL2(K)+Ch1/2Kv0vbL2(K))v0L2(K),

    where we have again used the trace inequality (4).

    Consequently,

    v0L2(K)wvNL2(K)+Ch1/2Kv0vbL2(K).

    Summing over all KTN yields

    εv022(εwvN2+CKTNεh1Kv0vb2L2(K)).

    Therefore, we have

    εv022(εwvN2+Csd(vN,vN)),

    which implies

    vNεCvNE. (3.5)

    From (3.4) and (3.5), we obtain the desired conclusion, which completes the proof.

    We shall show the coercivity of the bilinear form Aw(,) in E norm on V0N.

    Lemma 3.2. For any vNV0N, the following inequality holds:

    Aw(vN,vN)CvN2E,vNV0N. (3.6)

    Proof. For vN={v0,vb},wN={w0,wb}V0N, using the weak convection derivative (2.11) and integration by parts gives

    (bwvN,w0)=(v0,(bw0))+vb,bnw0=(bv0,w0)bn(v0vb),w0 (3.7)

    and

    (3.8)

    where we use the facts that , and in the last inequality.

    Combining (3.7) and (3.8), and taking , we obtain

    (3.9)

    From (3.9), we have

    which yields (3.6) with . The proof is completed.

    Therefore, the existence of a unique solution to (2.15) follows from the coercivity property of the bilinear form (3.6). As a result of the two lemmas above, the bilinear form is also coercive in the -norm in the sense that for any , there holds

    (3.10)

    Lemma 3.3. If for each , then there is a constant independent of and mesh size such that the solution defined in (2.15) satisfies

    (3.11)

    Proof. Choosing in (2.15) gives that

    Using the Cauchy-Schwarz inequality and the coercivity (3.6) of the bilinear form ,

    Integrating the above inequality with respect to the time variable , we arrive at

    (3.12)

    Using the Gronwall's inequality gives the desired conclusion. We are done.

    First, we define "vertices-edges' interpolation of a function on an element as follows. Let be the reference element with the vertices and the edges for . For , the approximation is given by

    (4.1)
    (4.2)
    (4.3)

    The approximation operator is well-defined [30]. Thus, we can define a continuous global interpolation operator by writing

    (4.4)

    where the bijective mapping is given by .

    This interpolation operator has the following stability estimate [30]

    (4.5)

    Since our approximation operator is continuous on , we have for . By the trace theorem, we will denote this by .

    Lemma 4.1. [31] For any ,

    We recall some technical results from [30].

    Lemma 4.2. For any and , there exists a constant such that the vertices-edges-element approximant satisfies

    for all .

    Lemma 4.3. Let . The following estimates hold for any :

    for all .

    A careful inspection of the proof of Lemma 4.3 in [30] reveals that the following results also hold true.

    Lemma 4.4. For and with , there exists a constant such that the vertices-edges-element approximant satisfies

    for all .

    Lemma 4.5. Let the assumption of Lemma 2.1 hold such that Then there is a constant such that the following interpolation error estimates are satisfied:

    (4.6)
    (4.7)
    (4.8)
    (4.9)
    (4.10)
    (4.11)

    Proof. The proof of (4.6) follows from Lemma 4.2 and the solution decomposition (2.2) of Lemma 2.1.

    Using the decay bound (2.3) of and the fact that , we have

    which shows (4.7) for . Similar arguments can be applied to the layer functions and . Thus, we complete the proof of (4.7). For (4.8) and (4.9), we will prove for since the other two parts follow similarly. One can use the decay bound (2.3) of to obtain

    We now shall prove (4.9). Appealing (2.3), one gets

    The proof of (4.10) is a little longer. Using an inverse estimate yields

    Hence, we shall estimate . With the help of the stability estimate (4.5) and the decay bound (2.3) of , we have

    If , then the sum can be small as a function of but not small if . For and , we have

    and when , again using the fact ,

    Thus,

    which proves (4.10). To prove (4.11), we use (4.7) and (4.10) to obtain

    On the set , from the triangle inequality, one obtains

    For , we have

    For , using (4.5) and the decay property (2.3) of ,

    (4.12)

    where we have used that Applying Lemma 4.2 and the decay property (2.3), we have for any

    where we have used and Lemma 2.2. Similarly, one can show that (4.11) holds for as well.

    For , one can prove as above . For , we obtain

    Thus, we complete the proof of (4.11). The proof of the lemma is now completed.

    Lemma 4.6. Let . Assume that the conditions of Lemma 4.5 hold. Then, we have

    for , where denotes , or .

    Proof. The first and second estimates follow from Lemma 4.3, Lemma 4.4, and the fact that . From the triangle inequality and (4.8) and (4.10) of Lemma 4.5, we have

    where we have used that . This completes the proof of the third and fifth inequalities for . An inverse inequality and (4.9) and (4.10) of Lemma 4.5 lead to

    where again we have used . This proves the third and fifth inequalities for . Using Lemma 4.4 with for and the decay bound (2.3) of , one can show that for any ,

    Similarly, one can prove that the result holds for , too.

    For , we get . For , we obtain

    which completes the proof of the fourth inequality. Similarly, one can prove that the last inequality holds true. The proof is now finished.

    Unlike the classical numerical methods such as FEM and the SUPG, the proposed WG-FEM does not have Galerkin orthogonality property. This results in some inconsistency errors in the error bounds. We first give a useful error equation in the following lemma.

    Lemma 5.1. [31] Let solve the problem (1.1). For ,

    (5.1)
    (5.2)
    (5.3)

    where

    (5.4)
    (5.5)
    (5.6)

    The following error equation will be needed in the error analysis.

    Lemma 5.2. Let and be the solutions of (1.1) and (2.15), respectively. For , one has

    (5.7)

    where , which are defined by (5.4), (5.5), and (5.6), respectively.

    Proof. Multiplying (1.1) by a test function , we arrive at

    With the help of (5.1), (5.2), and (5.3), the above equation becomes

    Since is continuous in , we get

    Therefore, we have

    (5.8)

    Subtracting (2.15) from (5.8) gives (5.7), which completes completed.

    Lemma 5.3. Let be the vertex-edge-cell interpolation of the solution of the problem (1.1). Then, there holds

    Proof. The solution decomposition (2.1) implies that

    Using Lemma 4.2 with and Lemma 2.1,

    Next, we examine the layer parts one by one. Let . From the stability property (4.5) of the interpolation operator, one has

    Let . The stability property (4.5) and Lemma 4.2 with yield

    Similarly, we can derive the estimates on the other layer components and . Combining the above estimates gives the desired conclusion.

    Thus, we complete the proof.

    We recall the following trace inequality. For any , one has

    (5.9)

    Lemma 5.4. Let and be given by (2.13). Assume that the conditions of Lemma 4.5 hold. Then, one has

    Proof. For the sake of simplicity, we use the following notations. Let and represent the interpolation errors of the regular and layer components of the solution. Hence, by the triangle inequality, we have

    (5.10)

    With the help of the trace inequality (5.9), we have

    Now, appealing the definition (2.13) of stabilization parameter and Lemma 4.6 gives

    (5.11)

    where we have used that .

    Using once again the trace inequality (5.9), we have

    Now, appealing the definition (2.13) of stabilization parameter and Lemma 4.6 again reveals that

    (5.12)

    Plugging (5.12) and (5.11) into (5.10) yields

    Consequently, we have

    which completes the proof.

    Now, we shall prove the error bounds for the consistency errors.

    Lemma 5.5. (A priori bounds) Assume that is the tensor product mesh as defined in Section 2. Then, for and we have

    (5.13)
    (5.14)

    Proof. It follows from the Cauchy-Schwarz and Holder inequalities that

    (5.15)

    Now, it then follows from Lemma 4.6 that

    (5.16)

    Next, we consider the term . From the Cauchy-Schwarz inequality and Lemma 5.4, we have

    (5.17)

    Combining (5.16) and (5.17), we get

    (5.18)

    From (5.5) and (5.6) and using , we arrive at

    Now, the Hölder inequality and Lemma 5.3 lead us to write

    (5.19)

    The Cauchy Schwartz and inverse inequalities give

    (5.20)

    Appealing the Cauchy Schwartz inequality on , we have

    (5.21)

    Using the error bounds (5.20) and (5.21) in (5.19), we obtain

    (5.22)

    where we have used the fact that .

    Since and are continuous, we conclude that for any . Then, the Hölder inequality and Lemma 5.3 imply that

    where we have used that and .

    Using the Cauchy Schwartz inequality and (4.6) and (4.11) of Lemma 4.5, we obtain

    The proof is completed.

    By letting in (5.7), we obtain

    It then follows from the estimates (5.13) and (5.14), together with Young's inequality and 3.10, that

    As a result,

    (5.23)

    Then, by integrating from to , we have

    This result is collected in the following theorem.

    Theorem 5.1. (Semi-discrete estimate) Let be the solution of (1.1)-(1.2) and be the solution of (2.15). Then, we have

    In this section, we shall use the Crank-Nicolson scheme on uniform time mesh in time to derive the fully discrete approximation of the problem (1.1) and (1.2). For a given partition of the time interval for some positive integer and step length , we define

    where the sequence . For simplicity, we denote by for a function We now state our fully discrete weak Galerkin finite element approximation. Find such that

    (6.1)

    with and .

    The following lemma shows that the Crank-Nicolson scheme is unconditionally stable in the norm.

    Lemma 6.1. Let . Then, we have the following stability estimate for the fully-discrete scheme (6.1):

    (6.2)

    Proof. Choosing in (6.1), and using the Cauchy-Schwarz inequality, we get

    where we have used that . Using the fact that and the coercivity of the bilinear form, we have

    Let be an integer. We sum the above inequality from to :

    Recalling that , the result follows. The proof is now completed.

    Next, we shall present the convergence analysis. To begin, we prove the error estimate of the discretization error . To this end, we need to derive an error equation involving the error .

    We formulate the error equation for in the following lemma.

    Lemma 6.2. For we have

    (6.3)

    where ; and

    Proof. From (1.1), one obtains the following equation:

    (6.4)

    On each element for we test equation (6.4) against to arrive at

    (6.5)

    Using a similar argument in deriving (5.8), one can show that

    where we have used that since is continuous in . Thus, we get

    (6.6)

    Subtracting (6.1) from (6.6) gives the conclusion. We complete the proof.

    Lemma 6.3. Let . Assume that and are the solutions (1.1), (1.2), and (6.1), respectively. One has for ,

    (6.7)

    Proof. Choosing in (6.3) and by the coercivity property (3.6), we find

    or, equivalently,

    (6.8)

    We can express the term . We write

    (6.9)

    and

    (6.10)

    From (6.9) and (6.10), we obtain

    (6.11)

    Hence, with the aid of the Cauchy-Schwarz and the Poincare inequality, in (6.8) can be estimated as follows.

    (6.12)

    where we have used the Young's inequities in the second inequality, and the estimate (6.11) and Lemma 4.5 in the second estimates of the righthand side. Applying Lemma 5.5 and Young's inequality, we obtain the estimate of the term in the righthand side of (6.8) as follows:

    (6.13)

    Combining (6.8)–(6.13) yields

    Let . Using the fact that , we sum the above expression from to to obtain

    We complete the proof.

    Theorem 6.1. Let . Assume that and are the solutions (1.1), (1.2), and (6.1), respectively. One has for ,

    Proof. Choosing in (6.3) and by coercivity (3.6), we find

    or, equivalently,

    Thus, we have

    Because , we sum up the above term from to for any fixed to get

    (6.14)

    From (6.11), we have

    (6.15)

    Observe that

    (6.16)

    Similar to (6.13), one has

    (6.17)

    It follows from Lemma 5.5, the Cauchy-Schwarz inequality, and Young's inequality that

    (6.18)

    From (6.16), (6.17), and (6.18) together with , we have

    (6.19)

    Combining (6.14), (6.15), and (6.19) yields that

    Finally, using (6.7), we obtain

    which completes the proof.

    This section presents various numerical examples for the fully-discrete Crank-Nicolson WG finite element method. We used MATLAB R2020A in our the calculations. We also used the 5-point Gauss-Legendre quadrature rule for evaluating of all integrals. All the calculations were calculated using MATLAB R2016a. The systems of linear equations resulting from the discrete problems were solved by lower-upper (LU) decomposition.

    We apply the fully-discrete WG-FEM on the adaptive meshes shown in Table 1. We choose and calculate the energy-norm and the -norm error , where is the error using intervals in each direction. The order of convergence (OC) is computed by the formula

    The numerical errors and the order of convergences in space are also tested. In order for the space error to dominate the errors, we take for element in each direction. We list the errors in the energy norm and -norm and the order of convergence in Tables 2 and 3, respectively. These numerical results show that the order of convergence is of order and of order in the energy and norms, respectively, which support the stated error estimates in Theorem 6.1.

    Table 2.  The energy-error and the order of convergence in space for Example 7.1 .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    16
    32 0.90 0.90 0.93
    64 0.93 0.93 0.93
    128 0.95 0.95 0.93
    256 0.97 0.97 0.97
    512 0.99 1.00 0.99
    16
    32 1.90 1.80 1.97
    64 1.94 1.90 1.93
    128 1.96 1.94 1.93
    256 1.97 1.99 1.97
    512 2.00 1.99 2.00
    16
    32 2.90 2.80 2.90
    64 2.94 2.92 2.93
    128 2.97 2.97 2.97
    256 2.98 2.98 2.99
    512 3.00 3.00 3.00

     | Show Table
    DownLoad: CSV
    Table 3.  The error and the order of convergence in space for Example 7.1 .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    16
    32 1.94 1.94 1.94
    64 1.97 1.97 1.97
    128 1.98 1.98 1.98
    256 1.99 1.99 1.99
    512 1.99 1.99 1.99
    16
    32 2.93 2.93 2.93
    64 2.97 2.97 2.97
    128 2.98 2.98 2.98
    256 2.99 2.99 2.99
    512 3.00 3.00 3.00
    16
    32 3.93 3.93 3.93
    64 3.94 3.94 3.94
    128 3.97 3.97 3.97
    256 3.99 3.99 3.99
    512 4.00 4.00 4.00

     | Show Table
    DownLoad: CSV

    Example 7.1. Let and in the problem (1.1). We choose and such that the exact solution is

    where .

    In Figure 2, we plot the numerical solutions of the WG-FEM using the element on the three layer-adapted meshes given in Figure 1 for and .

    Figure 2.  Numerical solution of Example 7.1 for using .

    We next present the temporal convergence rate for Example 7.1. In order for the temporal error to dominate the error, we take and , and use the element. We report the results in the -norm and the energy norm in Tables 4 and 5, respectively. We see that the order of convergence in time is of order , which verifies the theoretical estimate claimed in Theorem 6.1.

    Table 4.  The error and the order of convergence in time for Example 7.1 .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    1/2
    1/4 2.03 2.03 2.03
    1/8 1.99 1.99 1.99
    1/16 2.00 2.00 2.00

     | Show Table
    DownLoad: CSV
    Table 5.  The energy error and the order of convergence in time for Example 7.1 .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    1/2
    1/4 2.00 2.00 2.00
    1/8 1.99 1.99 1.99
    1/16 2.00 2.00 2.00

     | Show Table
    DownLoad: CSV

    Lastly, we test the robustness of the WG-FEM method with respect to the small parameter for Example 7.1. We take and use the element for the values of . The results are reported in Table 6. These results show that the WG-FEM is robust with respect to the perturbation parameter .

    Table 6.  The energy-error in space for Example 7.1 for the values of .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type

     | Show Table
    DownLoad: CSV

    The order of convergence via loglog plot in the energy norm and norm are plotted in Figures 3 and 4, respectively, for Example 7.2. We observe that the order of convergence of order and of order in the energy and norms, respectively, which support the stated error estimates in Theorem 6.1 as in Example 7.1. To test the temporal error, we choose and , and use the element. We present the results in the -norm and the energy norm in Tables 7 and 8, respectively. We see that the order of convergence in time is of order as claimed in Theorem 6.1.

    Figure 3.  The order of convergence in energy norm via loglog plot for Example 7.2 for using and elements.
    Figure 4.  The order of convergence in -norm via loglog plot for Example 7.2 for using and elements.
    Table 7.  The error and the order of convergence in time for Example 7.2 .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    1/2
    1/4 1.20 1.20 1.20
    1/8 1.70 1.70 1.70
    1/16 2.00 2.00 2.00

     | Show Table
    DownLoad: CSV
    Table 8.  The energy error and the order of convergence in time for Example 7.2 .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    1/2
    1/4 1.92 1.92 1.92
    1/8 2.01 2.01 2.01
    1/16 2.02 2.02 2.02

     | Show Table
    DownLoad: CSV

    Example 7.2. Let , , and in the problem (1.1). We take and such that the exact solution is

    We also test the WG-FEM for Example 7.2 for the robustness against . The results are presented in Table 9 for and the element for the values of . Again, one sees that the WG-FEM is the parameter-uniform method.

    Table 9.  The energy-error in space for Example 7.2 for the values of .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type

     | Show Table
    DownLoad: CSV

    In this paper, we present the Crack-Nicolson- WG-FEM applied to the singularly perturbed parabolic convection-dominated problems in 2D. We use the Crack-Nicolson scheme in time on uniform mesh and the WG-FEM in space on three layer-adapted meshes: Shishkin, Bakhvalov-Shishkin, and Bakhvalov meshes. We prove (almost) uniform error estimates of order in the energy norm and second order estimate in time. With the use of a special interpolation operator, the error analysis of the semi-discrete WG-FEM and the fully discrete WG-FEM have been carried out. Various numerical examples are conducted to validate the convergence rate of the proposed method.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors declare there is no conflicts of interest.



    [1] R. De Blasis, F. Petroni, Price Leadership and Volatility Linkages between Oil and Renewable Energy Firms during the COVID-19 Pandemic, Energies, 14 (2021), 2608. https://doi.org/10.3390/en14092608 doi: 10.3390/en14092608
    [2] S. Managi, T. Okimoto, Does the price of oil interact with clean energy prices in the stock market? Japan and the world economy, 27 (2013), 1–9. https://doi.org/10.1016/j.japwor.2013.03.003 doi: 10.1016/j.japwor.2013.03.003
    [3] C. Özdurak, Nexus between crude oil prices, clean energy investments, technology companies and energy democracy, Green Finance, 3 (2021), 337–350. https://doi.org/10.3934/GF.2021017 doi: 10.3934/GF.2021017
    [4] I. Henriques, P. Sadorsky, Oil prices and the stock prices of alternative energy companies, Energy Econ., 30 (2008), 998–1010. https://doi.org/10.1016/j.eneco.2007.11.001 doi: 10.1016/j.eneco.2007.11.001
    [5] J. C. Reboredo, Is there dependence and systemic risk between oil and renewable energy stock prices?, Energy Econ., 48 (2015), 32–45. https://doi.org/10.1016/j.eneco.2014.12.009 doi: 10.1016/j.eneco.2014.12.009
    [6] R. Bondia, S. Ghosh, K. Kanjilal, International crude oil prices and the stock prices of clean energy and technology companies: Evidence from non-linear cointegration tests with unknown structural breaks, Energy, 101 (2016), 558–565. https://doi.org/10.1016/j.energy.2016.02.031 doi: 10.1016/j.energy.2016.02.031
    [7] H. Zhang, G. Cai, D. Yang, The impact of oil price shocks on clean energy stocks: Fresh evidence from multi-scale perspective, Energy, 196 (2020), 117099. https://doi.org/10.1016/j.energy.2020.117099 doi: 10.1016/j.energy.2020.117099
    [8] R. Su, J. Du, F. Shahzad, X. Long, Unveiling the effect of mean and volatility spillover between the United States economic policy uncertainty and WTI crude oil price, Sustainability, 12 (2020), 6662. https://doi.org/10.3390/su12166662 doi: 10.3390/su12166662
    [9] Y. Jiang, G. Tian, B. Mo, Spillover and quantile linkage between oil price shocks and stock returns: new evidence from G7 countries, Financ. Innov., 6 (2020), 1–26. https://doi.org/10.1186/s40854-020-00208-y doi: 10.1186/s40854-020-00208-y
    [10] Z.-s. Ouyang, M.-t. Liu, S.-s. Huang, T. Yao, Does the source of oil price shocks matter for the systemic risk?, Energy Econ., 109 (2022), 105958. https://doi.org/10.1016/j.eneco.2022.105958 doi: 10.1016/j.eneco.2022.105958
    [11] T. Liu, S. Hamori, Spillovers to renewable energy stocks in the US and Europe: are they different? Energies, 13(2020), 3162. https://doi.org/10.3390/en13123162 doi: 10.3390/en13123162
    [12] Y. Ghabri, A. Ayadi, K. Guesmi, Fossil energy and clean energy stock markets under COVID-19 pandemic, Appl. Econ., 53 (2021), 4962–4974. https://doi.org/10.1080/00036846.2021.1912284 doi: 10.1080/00036846.2021.1912284
    [13] K. Tiwari, E. J. A. Abakah, D. Gabauer, R. A. Dwumfour, Dynamic spillover effects among green bond, renewable energy stocks and carbon markets during COVID-19 pandemic: Implications for hedging and investments strategies, Global Finance J., 51 (2022), 100692. https://doi.org/10.1016/j.gfj.2021.100692 doi: 10.1016/j.gfj.2021.100692
    [14] T. Liu, S. Hamori, Does investor sentiment affect clean energy stock? Evidence from TVP-VAR-based connectedness approach, Energies, 14 (2021), 3442. https://doi.org/10.3390/en14123442 doi: 10.3390/en14123442
    [15] M. Yahya, K. Kanjilal, A. Dutta, G. S. Uddin, S. Ghosh, Can clean energy stock price rule oil price? New evidences from a regime-switching model at first and second moments, Energy Econ., 95 (2021), 105116. https://doi.org/10.1016/j.eneco.2021.105116 doi: 10.1016/j.eneco.2021.105116
    [16] C. Y.-L. Hsiao, W. Lin, X. Wei, G. Yan, S. Li, N. Sheng, The impact of international oil prices on the stock price fluctuations of China's renewable energy enterprises, Energies, 12 (2019), 4630. https://doi.org/10.3390/en12244630 doi: 10.3390/en12244630
    [17] X. Lv, X. Dong, W. Dong, Oil prices and stock prices of clean energy: New evidence from Chinese subsectoral data, Emerg. Mark. Financ. Tr., 57 (2021), 1088–1102. https://doi.org/10.1080/1540496X.2019.1689810 doi: 10.1080/1540496X.2019.1689810
    [18] J. Zhu, Q. Song, D. Streimikiene, Multi-Time Scale Spillover Effect of International Oil Price Fluctuation on China's Stock Markets, Energies, 13(2020), 4641. https://doi.org/10.3390/en13184641 doi: 10.3390/en13184641
    [19] K. H. Liow, J. Song, X. Zhou, Volatility connectedness and market dependence across major financial markets in China economy, Quant. Finance Econ., 5 (2021), 397–420. https://doi.org/10.3934/QFE.2021018 doi: 10.3934/QFE.2021018
    [20] A. Dutta, Oil price uncertainty and clean energy stock returns: New evidence from crude oil volatility index, J. Clean. Prod., 164 (2017), 1157–1166. https://doi.org/10.1016/j.jclepro.2017.07.050 doi: 10.1016/j.jclepro.2017.07.050
    [21] Y. Song, Q. Ji, Y.-J. Du, J.-B. Geng, The dynamic dependence of fossil energy, investor sentiment and renewable energy stock markets, Energy Econ., 84 (2019), 104564. https://doi.org/10.1016/j.eneco.2019.104564 doi: 10.1016/j.eneco.2019.104564
    [22] K. Tiwari, S. Nasreen, S. Hammoudeh, R. Selmi, Dynamic dependence of oil, clean energy and the role of technology companies: New evidence from copulas with regime switching, Energy, 220 (2021), 119590. https://doi.org/10.1016/j.energy.2020.119590 doi: 10.1016/j.energy.2020.119590
    [23] M. Foglia, E. Angelini, Volatility connectedness between clean energy firms and crude oil in the COVID-19 era, Sustainability, 12 (2020), 9863. https://doi.org/10.3390/su12239863 doi: 10.3390/su12239863
    [24] Z. Li, Z. Huang, P. Failler, Dynamic correlation between crude oil price and investor sentiment in China: Heterogeneous and asymmetric effect, Energies, 15 (2022), 687. https://doi.org/10.3390/en15030687 doi: 10.3390/en15030687
    [25] P. Sadorsky, Correlations and volatility spillovers between oil prices and the stock prices of clean energy and technology companies, Energy econ., 34 (2012), 248–255. https://doi.org/10.1016/j.eneco.2011.03.006 doi: 10.1016/j.eneco.2011.03.006
    [26] W. Ahmad, P. Sadorsky, A. Sharma, Optimal hedge ratios for clean energy equities, Econ. Model., 72 (2018), 278–295. https://doi.org/10.1016/j.econmod.2018.02.008 doi: 10.1016/j.econmod.2018.02.008
    [27] I. Maghyereh, B. Awartani, H. Abdoh, The co-movement between oil and clean energy stocks: A wavelet-based analysis of horizon associations, Energy, 169 (2019), 895–913. https://doi.org/10.1016/j.energy.2018.12.039 doi: 10.1016/j.energy.2018.12.039
    [28] L. Pham, Do all clean energy stocks respond homogeneously to oil price?, Energy Econ., 81 (2019), 355–379. https://doi.org/10.1016/j.eneco.2019.04.010 doi: 10.1016/j.eneco.2019.04.010
    [29] N. Antonakakis, I. Chatziantoniou, D. Gabauer, Refined measures of dynamic connectedness based on time-varying parameter vector autoregressions, J. Risk Financ. Manag., 13 (2020), 84. https://doi.org/10.3390/jrfm13040084 doi: 10.3390/jrfm13040084
    [30] G. E. Primiceri, Time varying structural vector autoregressions and monetary policy, Rev. Econ. Stud., 72 (2005), 821–852. https://doi.org/10.1111/j.1467-937X.2005.00353.x doi: 10.1111/j.1467-937X.2005.00353.x
    [31] F. X. Diebold, K. Yılmaz, On the network topology of variance decompositions: Measuring the connectedness of financial firms, J. econometrics, 182 (2014), 119–134. https://doi.org/10.1016/j.jeconom.2014.04.012 doi: 10.1016/j.jeconom.2014.04.012
    [32] G. Koop, D. Korobilis, A new index of financial conditions, Eur. Econ. Rev., 71 (2014), 101–116. https://doi.org/10.1016/j.euroecorev.2014.07.002 doi: 10.1016/j.euroecorev.2014.07.002
    [33] R. Ferrer, S. J. H. Shahzad, R. López, F. Jareño, Time and frequency dynamics of connectedness between renewable energy stocks and crude oil prices, Energy Econ., 76 (2018), 1–20. https://doi.org/10.1016/j.eneco.2018.09.022 doi: 10.1016/j.eneco.2018.09.022
    [34] G. S. Uddin, M. L. Rahman, A. Hedström, A. Ahmed, Cross-quantilogram-based correlation and dependence between renewable energy stock and other asset classes, Energy Econ., 80 (2019), 743–759. https://doi.org/10.1016/j.eneco.2019.02.014 doi: 10.1016/j.eneco.2019.02.014
    [35] J. Peng, Z. Li, B. M. Drakeford, Dynamic characteristics of crude oil price fluctuation—from the perspective of crude oil price influence mechanism, Energies, 13 (2020), 4465. https://doi.org/10.3390/en13174465 doi: 10.3390/en13174465
    [36] M. A. Naeem, Z. Peng, M. T. Suleman, R. Nepal, S. J. H. Shahzad, Time and frequency connectedness among oil shocks, electricity and clean energy markets, Energy Econ., 91 (2020), 104914. https://doi.org/10.1016/j.eneco.2020.104914 doi: 10.1016/j.eneco.2020.104914
    [37] K. F. Kroner, V. K. Ng, Modeling asymmetric comovements of asset returns, Rev. Financ. Stud., 11 (1998), 817–844. https://doi.org/10.1093/rfs/11.4.817 doi: 10.1093/rfs/11.4.817
    [38] A. Timonina-Farkas, COVID-19: data-driven dynamic asset allocation in times of pandemic, Quant. Financ. Econ., 5 (2021), 198–227. https://doi.org/10.3934/QFE.2021009 doi: 10.3934/QFE.2021009
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2027) PDF downloads(84) Cited by(5)

Figures and Tables

Figures(10)  /  Tables(5)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog