
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 v∗1, 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 v∗1, 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=0on∂Ω, | (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 uh∈Vh and ph∈Wh such that
(μ∇wuh,∇wv)+(∇wph,v)=(f,v)∀v∈Vh, | (1.4) |
(uh,∇wq)=0∀q∈Wh, | (1.5) |
where ∇w denotes weak gradients to be defined in (2.3) and (2.4), and Vh and Wh are Pk and Pk−1 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|e∈Pk(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, Vh−Wh, 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 T∈Th, and we let mesh size h=maxT∈ThhT.
For k≥1 and given Th, the finite element space for velocity is defined by
Vh={v={v0,vb}: v0|T∈[Pk(T)]d, T∈Th;vb|e∈[Pk+1(e)]d, e∈Eh; vb|e=0, e∈Eh∩∂Ω}, | (2.1) |
and the finite element space for pressure is defined by
Wh={qh={q0,qb}: q0|T∈Pk−1(T), T∈T; qb|e∈Pk(e), e∈Eh;(q0,1)Th+⟨qb,1⟩∂Th=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=∑T∈Th(⋅,⋅)T and ⟨⋅,⋅⟩∂Th=∑T∈Th⟨⋅,⋅⟩∂T.
For a function v∈Vh, 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, τ⋅n⟩∂T∀τ∈[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 T∈Th, ∇wq∈[Pk(T)]d satisfies
(∇wq,φ)T=−(q0,∇⋅φ)T+⟨qb,φ⋅n⟩∂T∀φ∈[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 T∈Th. Let Πbk be a generic edge/face-wise defined L2 projection onto [Pk(e)]j for e∈∂T. Define Qhu={Πku,Πbk+1u}∈Vh and Qhp={Πk−1p,Πbkp}∈Wh.
Lemma 2.1. Let ϕ∈[H10(Ω)]d and ψ∈H1(Ω), then, for T∈Th
∇wQhϕ=Πk+1∇ϕ, | (2.5) |
∇wQhψ=Πk∇ψ. | (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,τ⋅n⟩∂T=−(ϕ,∇⋅τ)T+⟨ϕ,τ⋅n⟩∂T=(∇ϕ,τ)T=(Πk+1∇ϕ,τ)T. |
This implies that (2.5) holds. The scalar version, (2.6), is proved in the same manner.
For any function φ∈H1(T), the following trace inequality holds true:
‖φ‖2e≤C(h−1T‖φ‖2T+hT‖∇φ‖2T). | (2.7) |
We define two semi-norms |||v||| and ‖v‖1,h for any v∈Vh:
|||v|||2=(∇wv,∇wv), | (2.8) |
‖v‖21,h=∑T∈Th‖∇v0‖2T+∑T∈Thh−1T‖v0−vb‖2∂T. | (2.9) |
We also define two semi-norms |||q||| and ‖q‖1,h for any q∈Wh:
|||q|||2=(∇wq,∇wq), | (2.10) |
‖q‖21,h=∑T∈Th‖∇q0‖2T+∑T∈Thh−1T‖q0−qb‖2∂T. | (2.11) |
In fact, ‖v‖1,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:
C1‖v‖1,h≤|||v|||≤C2‖v‖1,h∀v∈Vh, | (2.12) |
and
C1‖q‖1,h≤|||q|||≤C2‖q‖1,h∀q∈Wh. | (2.13) |
Lemma 2.2. The following inf-sup conditions hold, for all q={q0,qb}∈Wh and v={v0,vb}∈Vh:
supv∈Vh(v0,∇wq)|||v|||≥β‖q0‖, | (2.14) |
and
supv∈Vh(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 ˜v∈H0(div,Ω) and ˜v|T∈[Pk(T)]d (see, e.g., [40, (7.4)–(7.6)]) such that
(∇⋅˜v,q0)|˜v|1,h≥C‖q0‖, | (2.16) |
where
|˜v|21,h=∑T∈Th(‖∇˜v‖2T+h−1T‖[˜v]‖2∂T). |
Let
v={˜v,˜vb}∈Vh,where ˜vb|e={12(˜v|T1+˜v|T2)if e∈E0h,0if e∈Eh∩∂Ω, |
where T1 and T2 are the two elements on the two sides of edge/triangle e. For such specially defined v, we have
‖v‖21,h=∑T∈Th(‖∇˜v‖2T+h−1T‖˜v−{˜v}‖2∂T)=∑T∈Th(‖∇˜v‖2T+h−1T‖[˜v]‖2∂T)=|˜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)=∑T∈Th(⟨˜v⋅n,qb⟩∂T−(∇⋅˜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)|‖˜v‖1,h≥β‖q0‖, |
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)T−⟨v0,τ⋅n⟩∂T≤‖∇v0‖T‖τ‖T+Ch−1/2T‖v0‖∂T‖τ‖T, |
which implies that
|||v|||≤Ch−1‖v0‖. | (2.20) |
It follows from (2.20) that for any v={v0,0}∈Vh
|(v0,∇wq)||||v|||≥Ch|(v0,∇wq)|‖v0‖. |
Then we have
supv∈Vh|(v0,∇wq)||||v|||≥Chsupv∈Vh|(v0,∇wq)|‖v0‖≥βh‖∇wq‖, |
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 ‖uh‖1,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 ‖ph‖1,h=0 and p0=pb=0 on ∂T. We have proved the lemma.
To derive the equations that the errors satisfy, we introduce eh=Qhu−uh and ϵh=Qhp−ph.
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) |
‖Πhτ−τ‖≤Chk+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 τ∈Hk+1(Ω). A quasi-projection πh can be defined such that πhτ∈H(div,Ω), πhτ|T∈[Pk(T)]d and for q0∈Pk−1(T),
(∇⋅τ,q0)T=(∇⋅πhτ,q0)T, | (3.4) |
−(∇⋅τ,q0)=(πhτ,∇wq), | (3.5) |
‖πhτ−τ‖≤Chk+1|τ|k+1, | (3.6) |
where |⋅|k+1 is the semi-Hk+1 Sobolev norm on the space.
Lemma 3.3. For any v∈Vh and q∈Wh, the following error equations hold true:
(μ∇weh,∇wv)+(∇wϵh,v0)=e1(u,v), | (3.7) |
(e0, ∇wq)=e2(u,q), | (3.8) |
where
e1(u, v)=μ(Πk+1∇u−Πh∇u,∇wv), | (3.9) |
e2(u,q)=(Πku−πhu,∇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
−(μ∇⋅∇u,v0)=(μΠh∇u,∇wv)=(μ∇wQhu,∇wv)−e1(u,v). | (3.12) |
It follows from (2.6) that
(∇p, v0)=(Πk∇p, v0)=(∇wQhp, v0). | (3.13) |
We substitute (3.12) and (3.13) into (3.11) to obtain
(μ∇wQhu,∇wv)+(μ∇wQhp, v0)=(f,v0)+e1(u,v). | (3.14) |
We subtract (3.14) from (1.4) to get
(μ∇weh,∇wv)+(μ∇wϵh, v0)=e1(u,v)∀v∈Vh. | (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, v∈Vh and q∈Wh. 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+1∇u−Πh∇u,∇wv)|≤Cμhk|u|k+1|||v|||. |
Similarly, we have
|e2(u, q)|=|(πhu−Πku,∇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:
|||Qhu−uh|||≤Chk|u|k+1, | (4.3) |
h|||Qhp−ph|||+‖Πk−1p−p0‖≤Cμhk|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)|=|(μ∇weh,∇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=|μ−1e1(u,eh)−e2(u,ϵh)|. |
It then follows from (4.1), (4.2) and (4.6) that
|||eh|||2≤Chk|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}=Qhu−uh and ϵh=Qhp−ph. Consider the problem of seeking (ψ,ξ) such that
−μΔψ+∇ξ=e0inΩ, | (4.7) |
∇⋅ψ=0inΩ, | (4.8) |
ψ=0on∂Ω. | (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:
μ‖ψ‖2+‖ξ‖1≤C‖e0‖. | (4.10) |
We need the following lemma first.
Lemma 4.2. For any v∈Vh and q∈Wh, the following equations hold true:
(μ∇wQhψ,∇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(∇ψ−Πk+1∇ψ)⋅n,v0−vb⟩∂Th,e4(ψ,q)=⟨(ψ−Πkψ)⋅n,q0−qb⟩∂Th. |
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 ⟨∇ψ⋅n,vb⟩∂Th=0, we derive
−(Δψ,v0)=(∇ψ,∇v0)Th−⟨∇ψ⋅n,v0−vb⟩∂Th. | (4.14) |
By integration by parts, and given(2.3) and (2.5), we have
(∇ψ,∇v0)Th=(Πk+1∇ψ,∇v0)Th=−(v0,∇⋅(Πk+1∇ψ))Th+⟨v0,Πk+1∇ψ⋅n⟩∂Th=(Πk+1∇ψ,∇wv)Th+⟨v0−vb,Πk+1∇ψ⋅n⟩∂Th=(∇wQhψ,∇wv)Th+⟨v0−vb,Πk+1∇ψ⋅n⟩∂Th. | (4.15) |
Combining (4.14) and (4.15) gives
−(μΔψ,v0)=(μ∇wQhψ,∇wv)−e3(ψ,v). | (4.16) |
By applying the definition of Πk, (2.6), and (3.13), we obtain
(∇ξ, v0)=(Πk∇ξ, 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+⟨ψ⋅n,q0−qb⟩∂Th=(∇⋅Πkψ,q0)Th−⟨Πkψ⋅n,q0⟩∂Th+⟨ψ⋅n,q0−qb⟩∂Th=−(Πkψ,∇wq)−⟨Πkψ⋅n,q0−qb⟩∂Th+⟨ψ⋅n,q0−qb⟩∂Th=−(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.,
(μ∇weh,∇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
(μ∇weh,∇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
(μ∇wQhψ,∇weh)+(∇wQhξ,e0)=(e0,e0)+e3(ψ,eh), | (4.25) |
(Πkψ,∇wϵh)=e4(ψ,ϵh), | (4.26) |
By applying (4.26), (4.23) becomes
(μ∇wQhψ,∇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:
‖Πku−u0‖≤Chk+1|u|k+1. | (4.28) |
Proof. Letting v=eh in (4.11) yields
‖e0‖2=(μ∇wQhψ,∇weh)Th−(e0,∇wQhξ)+e3(ψ,eh). | (4.29) |
By applying (4.27) and (4.24), (4.29) becomes
‖eh‖2=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|⟨(∇u−Πk+1∇u)⋅n,Πkψ−Πbk+1ψ⟩∂Th|≤μ(∑T∈Th‖∇u−Πk+1∇u‖2∂T)1/2(∑T∈Th‖Πkψ−ψ‖2∂T)1/2≤Cμhk+1|u|k+1|ψ|2. | (4.31) |
Similarly, we have
|e4(u,Qhξ)|≤|⟨(u−Πku)⋅n,Πk−1ξ−Πbkξ⟩∂Th|≤(∑T∈Th‖u−Πku‖2∂T)1/2(∑T∈Th‖Πk−1ξ−ξ‖2∂T)1/2≤Chk+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|⟨(∇ψ−Πk+1∇ψ)⋅n,e0−eb⟩∂Th|≤μ(∑T∈ThhT‖∇ψ−Πk+1∇ψ‖2∂T)1/2(∑T∈Thh−1T‖e0−eb‖2∂T)1/2≤Cμh|ψ|2|||eh|||≤Cμhk+1|u|k+1|ψ|2. | (4.33) |
By (4.4), we have
|e4(ψ,ϵh)|≤|⟨(ψ−Πkψ)⋅n,ϵ0−ϵb⟩∂Th|≤(∑T∈Th‖ψ−Πkψ‖2∂T)1/2(∑T∈Th‖ϵ0−ϵb‖2∂T)1/2≤Ch|ψ|2h|||ϵh|||≤Cμhk+1|u|k+1|ψ|2. | (4.34) |
Substituting all the four bounds above into (4.30), we get
‖eh‖2≤Chk+1|u|k+1(μ‖ψ‖2+‖ξ‖1). |
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=( (2y−6y2+4y3)(x2−2x3+x4)−(2x−6x2+4x3)(y2−2y3+y4)),p=−2x3+3x2−x. | (5.1) |
We computed the solution (5.1) on triangular grids shown, as in Figure 1 for the Pk-WG/Pk−1-WG mixed finite elements for k=1,2,3,4, and 5. The results are listed in Tables 1–5. 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.
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
We note that for some high level grids the computer round-off error was found to exceed the truncation error when μ=10−6, as described in Tables 3–5.
We computed the 2D solution (5.1) again for slightly perturbed triangular meshes, as illustrated in Figure 2 by employing the Pk-WG/Pk−1-WG mixed finite elements for k=1,2,3,4 and 5. The computational results are listed in Tables 6–10. The quasi-optimal convergence has been achieved for all solutions in all norms.
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
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(x−1)2x2(y−1)2y2(z−3z2+2z3)210(x−1)2x2(y−1)2y2(z−3z2+2z3)210[(x−3x2+2x3)(y2−y)2−(x2−x)2(y−3y2+2y3)](z2−z)2),p=−10(3y2−2y3−y). | (5.2) |
The 3D meshes are illustrated in Figure 3. The computational results are listed in Tables 11–13.
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−3 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−3 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−3 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 |
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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Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−3 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−3 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−3 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−6 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−3 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−3 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 |
Grid | ‖u−uh‖0 | O(hr) | |||u−uh||| | O(hr) | ‖Πk−1p−p0‖0 | 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=10−3 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 |