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Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting

  • Received: 01 February 2020 Revised: 01 July 2020 Published: 24 August 2020
  • Primary: 34D20, 34C23; Secondary: 37N25

  • The objective of the current paper is to investigate the dynamics of a new bioeconomic predator prey system with only predator's harvesting and Holling type Ⅲ response function. The system is equipped with an algebraic equation because of the economic revenue. We offer a detailed mathematical analysis of the proposed model to illustrate some of the significant results. The boundedness and positivity of solutions for the model are examined. Coexistence equilibria of the bioeconomic system have been thoroughly investigated and the behaviours of the model around them are described by means of qualitative theory of dynamical systems (such as local stability and Hopf bifurcation). The obtained results provide a useful platform to understand the role of the economic revenue v. We show that a positive equilibrium point is locally asymptotically stable when the profit v is less than a certain critical value v1, while a loss of stability by Hopf bifurcation can occur as the profit increases. It is evident from our study that the economic revenue has the capability of making the system stable (survival of all species). Finally, some numerical simulations have been carried out to substantiate the analytical findings.

    Citation: Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting[J]. Electronic Research Archive, 2021, 29(1): 1641-1660. doi: 10.3934/era.2020084

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  • The objective of the current paper is to investigate the dynamics of a new bioeconomic predator prey system with only predator's harvesting and Holling type Ⅲ response function. The system is equipped with an algebraic equation because of the economic revenue. We offer a detailed mathematical analysis of the proposed model to illustrate some of the significant results. The boundedness and positivity of solutions for the model are examined. Coexistence equilibria of the bioeconomic system have been thoroughly investigated and the behaviours of the model around them are described by means of qualitative theory of dynamical systems (such as local stability and Hopf bifurcation). The obtained results provide a useful platform to understand the role of the economic revenue v. We show that a positive equilibrium point is locally asymptotically stable when the profit v is less than a certain critical value v1, while a loss of stability by Hopf bifurcation can occur as the profit increases. It is evident from our study that the economic revenue has the capability of making the system stable (survival of all species). Finally, some numerical simulations have been carried out to substantiate the analytical findings.



    In this manuscript, we detail the development of a new stabilizer-free weak Galerkin (WG) finite element method of any polynomial order in 2D and 3D, on triangular and tetrahedral meshes respectively, for obtaining the solutions of the stationary Stokes equations: Find unknown functions u (velocity) and p (pressure) such that

    μΔu+p=finΩ, (1.1)
    u=0inΩ, (1.2)
    u=0onΩ, (1.3)

    where the viscosity μ>0, and the domain Ω is a polygon or a polyhedron in Rd(d=2,3).

    The new pressure-robust, stabilizer-free WG method has a new variational formulation. Find uhVh and phWh such that

    (μwuh,wv)+(wph,v)=(f,v)vVh, (1.4)
    (uh,wq)=0qWh, (1.5)

    where w denotes weak gradients to be defined in (2.3) and (2.4), and Vh and Wh are Pk and Pk1 WG finite element spaces, to be defined in (2.1) and (2.2), respectively. Some studies on the WG methods are as follows: the parabolic equation [1,2,3], with two-order superconvergence on triangular meshes [4], the convection-diffusion equation [5,6], 4th order problems [7,8], with the conforming discontinuous Galerkin formulation [9,10,11], the Navier-Stokes equations [12,13,14], with the discrete maximum principle [15], the second order elliptic equations [16,17], with the energy conservation [18], the Darcy flow [19,20], the Oseen equations [21], the Stokes equations with pressure-robustness [22,23,24], the Maxwell equations [25], the Cahn-Hilliard equation [26], the div-curl equations [27], with adaptive refinements [28,29], the biharmonic equation [30,31], the biharmonic equation with continuous finite elements[32,33], the Stokes equations with H(div) elements [34,35], with two-order superconvergence under the CDG formulation [36,37], and with two-order superconvergence for the Stokes equations [38].

    We note that the moment equation (1.1) is not tested by applying H(div,Ω) functions in (1.4), different from most other pressure-robust methods such as those detailed in [22,23,24], but is tested by using discontinuous polynomials. The pointwise divergence-free property (pressure-robustness) is achieved by introducing pressure face variables, i.e., ph={p0,pb} where pb|ePk(e) on every face edge/triangle e of a mesh Th (see (2.2) below). The method was previously applied before in hybridized discontinuous Galerkin methods in [39].

    It is shown that the WG finite element pair, VhWh, is inf-sup stable. The method is shown pressure robust, i.e., both errors of the velocity and the pressure are independent of the pressure p and the viscosity μ in (1.1). This is necessary when μ is small. However, very few methods can achieve this. Under the inf-sup stability, we shall prove quasi-optimal approximation for the velocity in an H1-norm and in the L2 norm. The quasi-optimal convergence for pressure in the L2 norm is also proved. The theory is numerically verified by applying varying degrees of WG finite elements to both triangular and tetrahedral meshes.

    Let Th be a mesh on the domain Ω, consisting of conforming shape-regular triangles or shape-regular tetrahedrons. Here we let hT be the element diameter of TTh, and we let mesh size h=maxTThhT.

    For k1 and given Th, the finite element space for velocity is defined by

    Vh={v={v0,vb}: v0|T[Pk(T)]d, TTh;vb|e[Pk+1(e)]d, eEh; vb|e=0, eEhΩ}, (2.1)

    and the finite element space for pressure is defined by

    Wh={qh={q0,qb}: q0|TPk1(T), TT; qb|ePk(e), eEh;(q0,1)Th+qb,1Th=0}, (2.2)

    where Pk(e) and Pk(T) denote the space of polynomials of degree k or less on the edge/triangle e and triangle/tetrahedron T respectively, (,)Th=TTh(,)T and ,Th=TTh,T.

    For a function vVh, the (k+1)-degree weak gradient wv is a piecewise polynomial on the mesh Th, wv|T[Pk+1(T)]d×d, such that

    (wv, τ)T=(v0, τ)T+vb, τnTτ[Pk+1(T)]d×d. (2.3)

    For a function q={q0,qb}Wh, its weak gradient wq is defined as a piecewise vector-valued polynomial such that for each TTh, wq[Pk(T)]d satisfies

    (wq,φ)T=(q0,φ)T+qb,φnTφ[Pk(T)]d. (2.4)

    We denote by Πk the local/element-wise L2-orthogonal projection onto [Pk(T)]j where j=1,d,d×d and TTh. Let Πbk be a generic edge/face-wise defined L2 projection onto [Pk(e)]j for eT. Define Qhu={Πku,Πbk+1u}Vh and Qhp={Πk1p,Πbkp}Wh.

    Lemma 2.1. Let ϕ[H10(Ω)]d and ψH1(Ω), then, for TTh

    wQhϕ=Πk+1ϕ, (2.5)
    wQhψ=Πkψ. (2.6)

    Proof. Using (2.3) and integration by parts, we obtain the following for any τ[Pk+1(T)]d×d:

    (wQhϕ,τ)T=(Πkϕ,τ)T+Πbk+1ϕ,τnT=(ϕ,τ)T+ϕ,τnT=(ϕ,τ)T=(Πk+1ϕ,τ)T.

    This implies that (2.5) holds. The scalar version, (2.6), is proved in the same manner.

    For any function φH1(T), the following trace inequality holds true:

    φ2eC(h1Tφ2T+hTφ2T). (2.7)

    We define two semi-norms |||v||| and v1,h for any vVh:

    |||v|||2=(wv,wv), (2.8)
    v21,h=TThv02T+TThh1Tv0vb2T. (2.9)

    We also define two semi-norms |||q||| and q1,h for any qWh:

    |||q|||2=(wq,wq), (2.10)
    q21,h=TThq02T+TThh1Tq0qb2T. (2.11)

    In fact, v1,h is a norm in Vh and |||||| is also a norm in Vh, as they have been proved in [4] by applying the norm equivalence as follows:

    C1v1,h|||v|||C2v1,hvVh, (2.12)

    and

    C1q1,h|||q|||C2q1,hqWh. (2.13)

    Lemma 2.2. The following inf-sup conditions hold, for all q={q0,qb}Wh and v={v0,vb}Vh:

    supvVh(v0,wq)|||v|||βq0, (2.14)

    and

    supvVh(v0,wq)|||v|||βh|||q|||, (2.15)

    where β>0 is independent of h and Th.

    Proof. For any given q={q0,qb}Wh, it is known that there exists a function ˜vH0(div,Ω) and ˜v|T[Pk(T)]d (see, e.g., [40, (7.4)–(7.6)]) such that

    (˜v,q0)|˜v|1,hCq0, (2.16)

    where

    |˜v|21,h=TTh(˜v2T+h1T[˜v]2T).

    Let

    v={˜v,˜vb}Vh,where ˜vb|e={12(˜v|T1+˜v|T2)if eE0h,0if eEhΩ,

    where T1 and T2 are the two elements on the two sides of edge/triangle e. For such specially defined v, we have

    v21,h=TTh(˜v2T+h1T˜v{˜v}2T)=TTh(˜v2T+h1T[˜v]2T)=|˜v|21,h. (2.17)

    It follows from (2.13) and (2.17) that

    |||v|||C|˜v|1,h. (2.18)

    By (2.4), we have

    (˜v,wq)=TTh(˜vn,qbT(˜v,q0)T)=(˜v,q0). (2.19)

    Combining (2.16), (2.19) and (2.18) implies that

    |(˜v,wq)||||v|||=|(˜v,q0)||||v||||(˜v,q0)|˜v1,hβq0,

    which implies (2.14).

    Next we shall derive (2.15). For any v={v0,0}Vh and τ[Pk+1(T)]d×d, we have the following by (2.3), (2.7) and the inverse inequalities:

    (wv,τ)T=(v0,τ)T=(v0,τ)Tv0,τnTv0TτT+Ch1/2Tv0TτT,

    which implies that

    |||v|||Ch1v0. (2.20)

    It follows from (2.20) that for any v={v0,0}Vh

    |(v0,wq)||||v|||Ch|(v0,wq)|v0.

    Then we have

    supvVh|(v0,wq)||||v|||ChsupvVh|(v0,wq)|v0βhwq,

    which implies (2.15). This completes the proof of the lemma.

    Lemma 2.3. There is a unique solution for the WG finite element equations (1.4) and (1.5).

    Proof. We only need to show that zero is the unique solution of (1.4) and (1.5) if f=0. We let f=0 and v=uh in (1.4) and q=ph in (1.5). By summing the two equations, we get

    (wuh, wuh)=0.

    It implies that wuh=0 on T. By (2.12), we also obtain that uh1,h=0. Thus, uh=0.

    Since uh=0 and f=0, (1.4) is reduced to (v0,wph)=0 for any v={v0,vb}Vh. Then the inf-sup conditions (2.15) and (2.14) imply that wph=0 and p0=0. By (2.13), we have that ph1,h=0 and p0=pb=0 on T. We have proved the lemma.

    To derive the equations that the errors satisfy, we introduce eh=Qhuuh and ϵh=Qhpph.

    Lemma 3.1. [41, Theorem 3.1] For τ[Hk+2(Ω)]d, a quasi-projection Πh can be defined such that Πhτ[H(div,Ω)]d, Πhτ|T[Pk+1(T)]d×d and for v0[Pk(T)]d,

    (τ,v0)T=(Πhτ,v0)T, (3.1)
    (τ,v0)=(Πhτ,wv), (3.2)
    ΠhττChk+2|τ|k+2, (3.3)

    where ||k+2 is the semi-Hk+2 Sobolev norm on the space.

    Lemma 3.2. [41, Theorem 3.1] Let τHk+1(Ω). A quasi-projection πh can be defined such that πhτH(div,Ω), πhτ|T[Pk(T)]d and for q0Pk1(T),

    (τ,q0)T=(πhτ,q0)T, (3.4)
    (τ,q0)=(πhτ,wq), (3.5)
    πhττChk+1|τ|k+1, (3.6)

    where ||k+1 is the semi-Hk+1 Sobolev norm on the space.

    Lemma 3.3. For any vVh and qWh, the following error equations hold true:

    (μweh,wv)+(wϵh,v0)=e1(u,v), (3.7)
    (e0, wq)=e2(u,q), (3.8)

    where

    e1(u, v)=μ(Πk+1uΠhu,wv), (3.9)
    e2(u,q)=(Πkuπhu,wq). (3.10)

    Proof. First, we test (1.1) by applying v0 with v={v0,vb}Vh to obtain

    (μΔu,v0)+(p, v0)=(f,v0). (3.11)

    It follows from (3.2) and (2.5) that

    (μu,v0)=(μΠhu,wv)=(μwQhu,wv)e1(u,v). (3.12)

    It follows from (2.6) that

    (p, v0)=(Πkp, v0)=(wQhp, v0). (3.13)

    We substitute (3.12) and (3.13) into (3.11) to obtain

    (μwQhu,wv)+(μwQhp, v0)=(f,v0)+e1(u,v). (3.14)

    We subtract (3.14) from (1.4) to get

    (μweh,wv)+(μwϵh, v0)=e1(u,v)vVh. (3.15)

    Multiplying (1.2) by q={q0,qb}Wh, by applying (3.2), it follows that

    0=(u, q0)=(πhu,wq)=(Πku,wq)+e2(u,q), (3.16)

    which implies that

    (Πku,wq)=e2(u,q). (3.17)

    The difference between (3.17) and (1.5) implies (3.8). We have proved the lemma.

    We shall first prove the optimal order error estimates of the |||||| norm for the velocity uh, and of the L2 norm for the pressure ph.

    Lemma 4.1. Let u[Hk+1(Ω)]d, vVh and qWh. The following estimates hold:

    |e1(u, v)|Cμhk|u|k+1|||v|||, (4.1)
    |e2(u, q)|Chk+1|u|k+1|||q|||, (4.2)

    where e1(,) and e2(,) have been defined in (3.9) and (3.10), respectively.

    Proof. By the Cauchy-Schwarz inequality, the definitions of Πk+1 and Πh, we compute

    |e1(u, v)|=μ|(Πk+1uΠhu,wv)|Cμhk|u|k+1|||v|||.

    Similarly, we have

    |e2(u, q)|=|(πhuΠku,wq)|Chk+1|u|k+1|||q|||.

    We have proved the lemma.

    Theorem 4.1. Let (u,p)([Hk+1(Ω)]d[H10(Ω)]d)×(Hk(Ω)L20(Ω)) be the solutions of (1.1)–(1.3). Let (uh,ph)Vh×Wh be the solutions of (1.4) and (1.5). Then, the following error estimates hold true:

    |||Qhuuh|||Chk|u|k+1, (4.3)
    h|||Qhpph|||+Πk1pp0Cμhk|u|k+1. (4.4)

    Proof. It follows from (3.7) that for any v={v0,vb}Vh, by (4.1), we have

    |(wϵh,v0)|=|(μweh,wv)e1(u,v)|Cμ(|||eh|||+hk|u|k+1)|||v|||. (4.5)

    Then applying the estimate (4.5) and (2.15) yields

    h|||ϵh|||Cμ(|||eh|||+hk|u|k+1). (4.6)

    By letting v=eh in (3.7) and q=ϵh in (3.8), and by using (3.8), we have

    |||eh|||2=|μ1e1(u,eh)e2(u,ϵh)|.

    It then follows from (4.1), (4.2) and (4.6) that

    |||eh|||2Chk|u|k+1|||eh|||+Chk|u|k+1h|||ϵh|||Chk|u|k+1|||eh|||+Ch2k|u|2k+1,

    which implies (4.3). The estimate (4.4) follows from (4.5), (4.3) and the inf-sup conditions (2.14) and (2.15). We have proved the theorem.

    We shall derive next the optimal-order convergence for velocity in the L2 norm by using the duality argument. Recall that eh={e0,eb}=Qhuuh and ϵh=Qhpph. Consider the problem of seeking (ψ,ξ) such that

    μΔψ+ξ=e0inΩ, (4.7)
    ψ=0inΩ, (4.8)
    ψ=0onΩ. (4.9)

    Assume that the duality problem given by (4.7)–(4.9) has the H2(Ω)×H1(Ω)-regularity property that the solution (ψ,ξ)H2(Ω)×H1(Ω) and the following a priori estimate holds true:

    μψ2+ξ1Ce0. (4.10)

    We need the following lemma first.

    Lemma 4.2. For any vVh and qWh, the following equations hold true:

    (μwQhψ,wv)+(wQhξ,v0)=(e0,v0)+e3(ψ,v), (4.11)
    (Πkψ,wq)=e4(ψ,q), (4.12)

    where ψ and ξ are defined in (4.7), and

    e3(ψ, v)=μ(ψΠk+1ψ)n,v0vbTh,e4(ψ,q)=(ψΠkψ)n,q0qbTh.

    Proof. Testing (4.7) by applying v0 with v={v0,vb}Vh gives

    (μΔψ,v0)+(ξ, v0)=(e0,v0). (4.13)

    By performing integration by parts, and setting ψn,vbTh=0, we derive

    (Δψ,v0)=(ψ,v0)Thψn,v0vbTh. (4.14)

    By integration by parts, and given(2.3) and (2.5), we have

    (ψ,v0)Th=(Πk+1ψ,v0)Th=(v0,(Πk+1ψ))Th+v0,Πk+1ψnTh=(Πk+1ψ,wv)Th+v0vb,Πk+1ψnTh=(wQhψ,wv)Th+v0vb,Πk+1ψnTh. (4.15)

    Combining (4.14) and (4.15) gives

    (μΔψ,v0)=(μwQhψ,wv)e3(ψ,v). (4.16)

    By applying the definition of Πk, (2.6), and (3.13), we obtain

    (ξ, v0)=(Πkξ, v0)=(wQhξ, v0). (4.17)

    Combining (4.16) and (4.17) with (4.13) yields (4.11).

    Testing (4.8) by applying q0 with q={q0,qb}Wh gives

    (ψ, q0)=0. (4.18)

    By applying integration by parts, we obtain

    (ψ, q0)=(Πkψ,q0)Th+ψn,q0qbTh=(Πkψ,q0)ThΠkψn,q0Th+ψn,q0qbTh=(Πkψ,wq)Πkψn,q0qbTh+ψn,q0qbTh=(Πkψ,wq)+e4(ψ,q). (4.19)

    Combining (4.18) and (4.19) yields

    (Πkψ,wq)=e4(ψ,q). (4.20)

    We have proved the lemma.

    By the same argument as that for (4.16), (3.7) has another form, i.e.,

    (μweh,wv)+(wϵh,v0)=e3(u,v), (4.21)
    (e0,wq)=e4(u,q). (4.22)

    Letting v=Qhψ and q=Qhξ in (4.21) and (4.22), we obtain

    (μweh,wQhψ)+(wϵh,Πkψ)=e3(u,Qhψ), (4.23)
    (e0,wQhξ)=e4(u,Qhξ). (4.24)

    Letting v=eh and q=ϵh in (4.11) and (4.12), we have

    (μwQhψ,weh)+(wQhξ,e0)=(e0,e0)+e3(ψ,eh), (4.25)
    (Πkψ,wϵh)=e4(ψ,ϵh), (4.26)

    By applying (4.26), (4.23) becomes

    (μwQhψ,weh)=e3(u,Qhψ)+e4(ψ,ϵh). (4.27)

    Theorem 4.2. Let (u,p)([Hk+1(Ω)]d[H10(Ω)]d)×(Hk(Ω)L20(Ω)) be the solutions of (1.1)–(1.3). Let (uh,ph)Vh×Wh denote the unique solutions of (1.4) and (1.5). With the condition (4.10), the following error bound holds:

    Πkuu0Chk+1|u|k+1. (4.28)

    Proof. Letting v=eh in (4.11) yields

    e02=(μwQhψ,weh)Th(e0,wQhξ)+e3(ψ,eh). (4.29)

    By applying (4.27) and (4.24), (4.29) becomes

    eh2=e3(u,Qhψ)+e4(ψ,ϵh)+e3(ψ,eh)+e4(u,Qhξ). (4.30)

    Next we shall estimate all of the terms on the right hand side of (4.30). Using the Cauchy-Schwarz inequality, the trace inequality (2.7), and the definition of Πk+1, we obtain

    |e3(u,Qhψ)|μ|(uΠk+1u)n,ΠkψΠbk+1ψTh|μ(TThuΠk+1u2T)1/2(TThΠkψψ2T)1/2Cμhk+1|u|k+1|ψ|2. (4.31)

    Similarly, we have

    |e4(u,Qhξ)||(uΠku)n,Πk1ξΠbkξTh|(TThuΠku2T)1/2(TThΠk1ξξ2T)1/2Chk+1|u|k+1|ξ|1. (4.32)

    Using the Cauchy-Schwarz inequality and the trace inequalities, applying (2.12) and (4.3), we obtain

    |e3(ψ,eh)|μ|(ψΠk+1ψ)n,e0ebTh|μ(TThhTψΠk+1ψ2T)1/2(TThh1Te0eb2T)1/2Cμh|ψ|2|||eh|||Cμhk+1|u|k+1|ψ|2. (4.33)

    By (4.4), we have

    |e4(ψ,ϵh)||(ψΠkψ)n,ϵ0ϵbTh|(TThψΠkψ2T)1/2(TThϵ0ϵb2T)1/2Ch|ψ|2h|||ϵh|||Cμhk+1|u|k+1|ψ|2. (4.34)

    Substituting all the four bounds above into (4.30), we get

    eh2Chk+1|u|k+1(μψ2+ξ1).

    By applying this inequality and the regularity condition (4.10), (4.28) is proved.

    In the first example in 2D, we have chosen the domain Ω=(0,1)×(0,1) for the Stokes equations (1.1)–(1.3). We have chosen an f (depending on μ) in (1.1) such that the exact solution of (1.1)–(1.3) is as follows (independent of μ):

    u=( (2y6y2+4y3)(x22x3+x4)(2x6x2+4x3)(y22y3+y4)),p=2x3+3x2x. (5.1)

    We computed the solution (5.1) on triangular grids shown, as in Figure 1 for the Pk-WG/Pk1-WG mixed finite elements for k=1,2,3,4, and 5. The results are listed in Tables 15. As shown, the optimal order of convergence has been achieved for all solutions in all norms. From the data, we can see the method is pressure-robust and the error is independent of the viscosity μ.

    Figure 1.  The uniform triangular meshes used to compute the results in Tables 15.
    Table 1.  The order of convergence and the error results given by the P1/P0 WG element for the solution (5.1) for the Figure 1 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P1-WG/P0-WG elements, μ=1 in (1.1).
    5 0.5113E-03 1.94 0.2717E-01 0.99 0.1927E-01 1.00
    6 0.1285E-03 1.99 0.1357E-01 1.00 0.9706E-02 0.99
    7 0.3209E-04 2.00 0.6769E-02 1.00 0.4871E-02 0.99
    By the P1-WG/P0-WG elements, μ=106 in (1.1).
    5 0.5113E-03 1.94 0.2717E-01 0.99 0.1929E-07 1.00
    6 0.1285E-03 1.99 0.1357E-01 1.00 0.9700E-08 0.99
    7 0.3209E-04 2.00 0.6769E-02 1.00 0.4881E-08 0.99

     | Show Table
    DownLoad: CSV
    Table 2.  The order of convergence and the error results given by the P2/P1 WG element for the solution (5.1) for the Figure 1 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P2-WG/P1-WG elements, μ=1 in (1.1).
    4 0.9758E-04 3.03 0.6913E-02 1.91 0.1433E-01 1.13
    5 0.1185E-04 3.04 0.1747E-02 1.98 0.4625E-02 1.63
    6 0.1458E-05 3.02 0.4356E-03 2.00 0.1301E-02 1.83
    By the P2-WG/P1-WG elements, μ=106 in (1.1).
    4 0.9758E-04 3.03 0.6913E-02 1.91 0.1435E-07 1.13
    5 0.1185E-04 3.04 0.1747E-02 1.98 0.4643E-08 1.63
    6 0.1458E-05 3.02 0.4356E-03 2.00 0.1345E-08 1.79

     | Show Table
    DownLoad: CSV
    Table 3.  The order of convergence and the error results given by the P3/P2 WG element for the solution (5.1) for the Figure 1 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P3-WG/P2-WG elements, μ=1 in (1.1).
    4 0.6255E-05 3.92 0.5836E-03 2.89 0.1448E-02 2.58
    5 0.3879E-06 4.01 0.7294E-04 3.00 0.2066E-03 2.81
    6 0.2392E-07 4.02 0.9022E-05 3.02 0.2751E-04 2.91
    By the P3-wG/P2-WG elements, μ=106 in (1.1).
    4 0.6255E-05 3.92 0.5836E-03 2.89 0.1489E-08 2.54
    5 0.3879E-06 4.01 0.7294E-04 3.00 0.3668E-09 2.02
    6 0.2392E-07 4.02 0.9022E-05 3.02 0.3366E-09 0.12

     | Show Table
    DownLoad: CSV
    Table 4.  The order of convergence and the error results given by the P4/P3 WG element for the solution (5.1) for the Figure 1 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P4-WG/P3-WG elements, μ=1 in (1.1).
    3 0.1291E-04 4.63 0.6964E-03 3.67 0.6446E-03 3.47
    4 0.4222E-06 4.93 0.4501E-04 3.95 0.4543E-04 3.83
    5 0.1321E-07 5.00 0.2814E-05 4.00 0.2983E-05 3.93
    By the P4-WG/P3-WG elements, μ=106 in (1.1).
    3 0.1291E-04 4.63 0.6964E-03 3.67 0.8212E-09 3.12
    4 0.4222E-06 4.93 0.4501E-04 3.95 0.4223E-09 0.96
    5 0.1321E-07 5.00 0.2814E-05 4.00 0.4401E-09 0.00

     | Show Table
    DownLoad: CSV
    Table 5.  The order of convergence and the error results given by the P5/P4 WG element for the solution (5.1) for the Figure 1 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P5-WG/P4-WG elements, μ=1 in (1.1).
    2 0.5642E-04 4.37 0.1814E-02 3.60 0.9773E-03 3.55
    3 0.9738E-06 5.86 0.6161E-04 4.88 0.3342E-04 4.87
    4 0.1546E-07 5.98 0.1952E-05 4.98 0.1069E-05 4.97
    By the P5-WG/P4-WG elements, μ=106 in (1.1).
    2 0.5642E-04 4.37 0.1814E-02 3.60 0.1211E-08 3.24
    3 0.9738E-06 5.86 0.6161E-04 4.88 0.3831E-09 1.66
    4 0.1546E-07 5.98 0.1953E-05 4.98 0.5148E-09 0.00

     | Show Table
    DownLoad: CSV

    We note that for some high level grids the computer round-off error was found to exceed the truncation error when μ=106, as described in Tables 35.

    We computed the 2D solution (5.1) again for slightly perturbed triangular meshes, as illustrated in Figure 2 by employing the Pk-WG/Pk1-WG mixed finite elements for k=1,2,3,4 and 5. The computational results are listed in Tables 610. The quasi-optimal convergence has been achieved for all solutions in all norms.

    Figure 2.  The triangular meshes used to compute the results in Tables 610.
    Table 6.  The order of convergence and the error results given by the P1/P0 WG element for the solution (5.1) for the Figure 2 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P1-WG/P0-WG elements, μ=1 in (1.1).
    4 0.7627E-03 1.85 0.3264E-01 0.96 0.1809E-01 1.28
    5 0.1954E-03 1.96 0.1639E-01 0.99 0.7991E-02 1.18
    6 0.4904E-04 1.99 0.8189E-02 1.00 0.3835E-02 1.06
    By the P1-WG/P0-WG elements, μ=106 in (1.1).
    4 0.7627E-03 1.85 0.3264E-01 0.96 0.1808E-07 1.28
    5 0.1954E-03 1.96 0.1639E-01 0.99 0.7999E-08 1.18
    6 0.4904E-04 1.99 0.8189E-02 1.00 0.3846E-08 1.06

     | Show Table
    DownLoad: CSV
    Table 7.  The order of convergence and the error results given by the P2/P1 WG element for the solution (5.1) for the Figure 2 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P2-WG/P1-WG elements, μ=1 in (1.1).
    4 0.2225E-04 2.95 0.2167E-02 1.94 0.2011E-02 1.04
    5 0.2784E-05 3.00 0.5434E-03 2.00 0.6969E-03 1.53
    6 0.3480E-06 3.00 0.1355E-03 2.00 0.2017E-03 1.79
    By the P2-WG/P1-WG elements, μ=106 in (1.1).
    4 0.2225E-04 2.95 0.2167E-02 1.94 0.2031E-08 1.03
    5 0.2784E-05 3.00 0.5434E-03 2.00 0.7851E-09 1.37
    6 0.3480E-06 3.00 0.1355E-03 2.00 0.3701E-09 1.08

     | Show Table
    DownLoad: CSV
    Table 8.  The order of convergence and the error results given by the P3/P2 WG element for the solution (5.1) for the Figure 2 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P3-WG/P2-WG elements, μ=1 in (1.1).
    3 0.1339E-04 3.62 0.9788E-03 2.69 0.2254E-02 1.84
    4 0.8544E-06 3.97 0.1239E-03 2.98 0.3795E-03 2.57
    5 0.5341E-07 4.00 0.1541E-04 3.01 0.5423E-04 2.81
    By the P3-WG/P2-WG elements, μ=106 in (1.1).
    3 0.1339E-04 3.62 0.9788E-03 2.69 0.2311E-08 1.80
    4 0.8544E-06 3.97 0.1239E-03 2.98 0.4983E-09 2.21
    5 0.5341E-07 4.00 0.1541E-04 3.01 0.3679E-09 0.44

     | Show Table
    DownLoad: CSV
    Table 9.  The order of convergence and the error results given by the P4/P3 WG element for the solution (5.1) for the Figure 2 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P4-WG/P3-WG elements, μ=1 in (1.1).
    2 0.2795E-04 3.98 0.1327E-02 3.12 0.3037E-02 2.70
    3 0.8864E-06 4.98 0.8282E-04 4.00 0.2700E-03 3.49
    4 0.2742E-07 5.01 0.5105E-05 4.02 0.1973E-04 3.77
    By the P4-WG/P3-WG elements, μ=106 in (1.1).
    2 0.2795E-04 3.98 0.1327E-02 3.12 0.3134E-08 2.66
    3 0.8864E-06 4.98 0.8282E-04 4.00 0.5525E-09 2.50
    4 0.2742E-07 5.01 0.5105E-05 4.02 0.4121E-09 0.42

     | Show Table
    DownLoad: CSV
    Table 10.  The order of convergence and the error results given by the P5/P4 WG element for the solution (5.1) for the Figure 2 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P5-WG/P4-WG elements, μ=1 in (1.1).
    2 0.2375E-05 6.00 0.1322E-03 5.02 0.2879E-03 4.38
    3 0.3664E-07 6.02 0.4074E-05 5.02 0.1069E-04 4.75
    4 0.1017E-08 5.17 0.4051E-06 3.33 0.3607E-06 4.89
    By the P5-WG/P4-WG elements, μ=106 in (1.1).
    2 0.2375E-05 6.00 0.1322E-03 5.02 0.6884E-09 3.14
    3 0.3664E-07 6.02 0.4075E-05 5.02 0.4504E-09 0.61
    4 0.1013E-08 5.18 0.4030E-06 3.34 0.5125E-09 0.00

     | Show Table
    DownLoad: CSV

    In the third test, we performed 3D numerical computation on domain Ω=(0,1)×(0,1)×(0,1). We chose an f in (1.1) such that we would have the following exact solution

    u=(210(x1)2x2(y1)2y2(z3z2+2z3)210(x1)2x2(y1)2y2(z3z2+2z3)210[(x3x2+2x3)(y2y)2(x2x)2(y3y2+2y3)](z2z)2),p=10(3y22y3y). (5.2)

    The 3D meshes are illustrated in Figure 3. The computational results are listed in Tables 1113.

    Figure 3.  The tetrahedral meshes used to compute the results in Tables 1113.
    Table 11.  The order of convergence and the error results given by the P1/P0 WG finite element for the solution (5.2) for the Figure 3 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P1-WG/P0-WG elements, μ=1 in (1.1).
    3 0.366E-01 1.20 0.696E+00 0.73 0.412E+00 2.62
    4 0.975E-02 1.91 0.355E+00 0.97 0.812E-01 2.34
    5 0.242E-02 2.01 0.180E+00 0.98 0.145E-01 2.49
    By the P1-WG/P0-WG elements, μ=103 in (1.1).
    3 0.428E-01 1.03 0.763E+00 0.67 0.553E-03 2.42
    4 0.108E-01 1.98 0.362E+00 1.08 0.134E-03 2.05
    5 0.249E-02 2.12 0.180E+00 1.00 0.196E-04 2.77

     | Show Table
    DownLoad: CSV
    Table 12.  The order of convergence and the error results given by the P2/P1 WG finite element for the solution (5.2) for the Figure 3 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P2-WG/P1-WG elements, μ=1 in (1.1).
    3 0.618E-02 2.90 0.231E+00 1.83 0.145E+00 1.78
    4 0.639E-03 3.27 0.498E-01 2.21 0.258E-01 2.49
    5 0.476E-04 3.75 0.117E-01 2.09 0.540E-03 5.58
    By the P2-WG/P1-WG elements, μ=103 in (1.1).
    3 0.615E-02 2.91 0.209E+00 1.97 0.135E-03 1.91
    4 0.685E-03 3.17 0.473E-01 2.14 0.232E-04 2.54
    5 0.474E-04 3.85 0.117E-01 2.02 0.716E-06 5.02

     | Show Table
    DownLoad: CSV
    Table 13.  The order of convergence and the error results given by the P3/P2 WG finite element for the solution (5.2) for the Figure 3 meshes.
    Grid uuh0 O(hr) |||uuh||| O(hr) Πk1pp00 O(hr)
    By the P3-WG/P2-WG elements, μ=1 in (1.1).
    3 0.644E-03 3.89 0.376E-01 2.80 0.257E-01 3.70
    4 0.351E-04 4.20 0.480E-02 2.97 0.896E-03 4.84
    5 0.198E-05 4.15 0.618E-03 2.96 0.465E-04 4.27
    By the P3-WG/P2-WG elements, μ=103 in (1.1).
    3 0.708E-03 3.71 0.382E-01 2.70 0.314E-04 3.58
    4 0.507E-04 3.80 0.484E-02 2.98 0.140E-05 4.48
    5 0.210E-05 4.60 0.618E-03 2.97 0.398E-07 5.14

     | Show Table
    DownLoad: CSV

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

    Yan Yang is supported in part by the Program of Sichuan National Applied Mathematics Center, No. 2023-KFJJ-01-001.

    The authors declare there are no conflicts of interest.



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