The semidiscrete and fully discrete discontinuous mixed covolume schemes for the linear parabolic integrodifferential problems on triangular meshes are proposed. The error analysis of the semidiscrete and fully discrete discontinuous mixed covolume scheme is presented and the optimal order error estimate in discontinuous H(div) and first-order error estimate in L2 are obtained with the lowest order Raviart-Thomas mixed element space.
1. Introduction
We consider the following linear parabolic integrodifferential problems
(1)qt(x,t)-∇·{A(x,t)∇q(x,t)+∫0tB(x,t,τ)∇q(x,t,τ)dτ}=f(x,t),(x,t)∈Ω×(0,T],q(x,0)=q0(x),x∈Ω,q(x,t)=0,(x,t)∈∂Ω×(0,T],
where Ω∈R2 is a bounded convex polygonal domain with the boundary ∂Ω, x=(x,y), q is an unknown function, A is a symmetric, bounded matrix function, B is a bounded matrix function, q0(x) and f(x,t) are known functions, and f(x,t)∈L2(Ω). Furthermore, we assume that the matrix M=A-1 and Mt are two bounded matrix functions.
Here and in what follows, we will not write the independent x, t for any functions unless it is necessary.
For the parabolic integrodifferential problems many numerical methods were proposed, such as the finite element methods in [1], H1-Galerkin mixed finite element methods in [2], finite element approximation with a weakly singular kernel in [3], expanded mixed finite element methods in [4], and expanded mixed covolume method in [5].
Because the discontinuous Galerkin method has the advantages of a high order of accuracy, high parallelizability, localizability, and easy handling of complicated geometries, it has been used to solve elliptic problems and convection-diffusion problems by many researchers; see [6–11]. The discontinuous finite volume method in recent years was used to solve elliptic problems, Stokes problems, and parabolic problems in [12–14]. In [15] the discontinuous mixed covolume methods for elliptic problems were demonstrated by Yang and Jiang. Zhu and Jiang extended the discontinuous mixed covolume methods to parabolic problems in [16]. The goal of this paper is to extend the discontinuous mixed covolume methods in the linear parabolic integrodifferential problems.
The rest of this paper is organized as follows. In Section 2, some notations are introduced and the semidiscrete and the fully discrete discontinuous mixed covolume schemes for the integrodifferential equations (1) are established. In Section 3, the existence and uniqueness for the semidiscrete and the fully discrete discontinuous mixed covolume approximations are proven. We defined a generalized discontinuous mixed covolume elliptic projection in Section 4. We prove the optimal error estimations in both H1 and L2 norms of semidiscrete and the fully discrete discontinuous mixed covolume methods in Sections 5 and 6.
Throughout this paper, the letter C denotes a generic positive constant independent of the mesh parameter and may stand for different values at its different appearances.
2. Discontinuous Mixed Covolume Formulation
Let w=-A∇q and D=BM, and rewrite (1) as the system of first-order partial differential equations
(2)Mw+∇q=0,(x,t)∈Ω×(0,T],qt+∇·w+∫0t∇·(Dw)dτ=f,(x,t)∈Ω×(0,T],q(x,0)=q0(x),x∈Ω,q(x,t)=0,(x,t)∈∂Ω×(0,T].
We will use the standard definitions for the Sobolev spaces Hs(K) and their associated inner products (·,·)s,K, norms ∥·∥s,K and seminorms |·|s,K in [17]. The space H0(K) coincides with Ls(K), in which the norm and the inner product are denoted by ∥·∥K and (·,·)K, respectively.
Let Th={K} be a triangulation of the domain Ω. As usual, we assume the triangles K to be shape-regular. For a given triangulation Th, we construct a dual mesh Th* based upon the primal partition Th. Each triangle in Th can be divided into three subtriangles by connecting the barycenter Q of the triangle to their corner nodes Ai(i=1,2,3). Then we define the dual partition Th* to be the union of the triangles shown in Figure 1. Let Pk(T) consist of all the polynomials functions of degree less than or equal to k defined on T. We define the finite-dimensional trial function space for velocity on Th by
(3)Uh≔{u∈L2(Ω)2:u|K∈P1(K)2,∀K∈Th}.
Define the finite-dimensional test function space Vh for velocity associated with the dual partition Th* as
(4)Vh≔{v∈L2(Ω)2:v|T∈P0(T)2,∀T∈Th*}.
Let Hh be the finite-dimensional space for pressure
(5)Hh≔{p∈L2(Ω):p|K∈P0(K),∀K∈Th}.
Element T∈Th* for a triangular mesh.
Let Γ denote the union of the boundary of the triangles K of Th and Γ0:Γ∖∂Ω. The traces of functions in Vh and Hh are double valued on Γ0. Let e be an interior edge shared by two triangles K1 and K2 in Th. Define the normal vectors n1 and n2 on e pointing exterior to K1 and K2, respectively. Next, we introduce some traces operators that we will use in our numerical formulation. We define the average {·} and jump [·] on e for scalar q and vector v, respectively:
(6){p}=12(p|∂K1+p|∂K2),[p]=p|∂K1n1+p|∂K2n2,{u}=12(u|∂K1+u|∂K2),[u]=u|∂K1·n1+u|∂K2·n2;
if e is an edge on the boundary of Ω, we set
(7){p}=p,[u]=u·n,
where n is the outward unit normal. We do not require either of the quantities [p] or {u} on boundary edges, and we leave them undefined.
Multiplying the first and second equations in system (2) by v∈Vh and p∈Hh, respectively, and using the integration by parts formula in the equation, we have
(8)∑T∈Th*∫TMw·vdx+∑T∈Th*∫∂Tqv·nds=0,∀v∈Vh,∑K∈Th∫Kqtpdx+∑K∈Th∫K∇·wpdx+∑K∈Th∫K∫0t∇·(Dw)pdτdx=(f,p),∀p∈Hh,
where n is the outward normal vector on ∂T. Let Tj∈Th*(j=1,2,3) be the triangles in K∈Th. Then we have
(9)∑T∈Th*∫∂Tqv·nds=∑K∈Th∑j=13∫Aj+1QAjqv·nds+∑K∈Th∫∂Kqv·nds,∀v∈Vh,
where A4=A1. A straightforward computation gives
(10)∑K∈Th∫∂Kqv·nds=∑e∈Γ0∫e[q]·{v}ds+∑e∈Γ∫e{q}·[v]ds,∀v∈Vh.
Let ∫Γqds=∑e∈Γ∫eqds. Using the above formula and the fact that [q]=0 for q∈H1(Ω) on Γ0, (9) becomes
(11)∑T∈Th*∫∂Tqv·nds=∑K∈Th∑j=13∫Aj+1QAjqv·nds+∑e∈Γ∫e{q}[v]ds,∀v∈Vh.
Then, system (8) can be rewritten as in the following:
(12)∑T∈Th*∫TMw·vdx+∑K∈Th∑j=13∫Aj+1QAjqv·nds+∑e∈Γ∫e{q}[v]ds=0,∀v∈Vh,∑K∈Th∫Kqtpdx+∑K∈Th∫K∇·wpdx+∑K∈Th∫K∫0t∇·(Dw)pdτdx=(f,p),∀p∈Hh.
Let U(h)=Uh+H2(Ω)2. Define a mapping γ:U(h)→Vh as
(13)γu|T=1he∫eu|Tds,T∈Th*,
where he is the length of the edge e. For u=(u1,u2)∈U(h), γui(i=1,2) is defined as
(14)γui|T=1he∫eui|Tds,T∈Th*,(i=1,2).
Then the system (12) is equivalent to
(15)∑T∈Th*∫TMw·γvdx+∑K∈Th∑j=13∫Aj+1QAjqγv·nds+∑e∈Γ∫e{q}[γv]ds=0,∀v∈U(h),∑K∈Th∫Kqtpdx+∑K∈Th∫K∇·wpdx+∑K∈Th∫K∫0t∇·(Dw)pdτdx=(f,p),∀p∈Hh.
Let
(16)a0(u,v)≔∑T∈Th*∫TMu·γvdx,b(v,p)≔∑K∈Th∑j=13∫Aj+1QAjpγv·nds+∑e∈Γ∫e{p}[γv]ds,c0(u,p)≔∑K∈Th∫K∇·updx,c~0(u,p)≔∑K∈Th∫K∇·(Du)pdx.
Using the above bilinear forms, it is clear that system (15) can be rewritten as in the following:
(17)a0(w,v)+b(v,q)=0,∀v∈U(h),(qt,p)+c0(w,p)+∫0tc~0(w,p)dτ=(f,p),∀p∈Hh.
In order to define our numerical schemes, we introduce the bilinear forms as follows:
(18)a(u,v)≔a0(u,v)+α∑e∈Γ1he∫eM[u][v]ds,c(u,p)≔c0(u,p)-∫Γ{p}[γu]ds,c~(u,p)≔c~0(u,p)-∫Γ{p}[γu]ds,
where α>0 is a parameter to be determined later. For the exact solution (w,q) of system (2), we have
(19)a0(w,u)=a(w,u),∀u∈Uh,c0(w,p)=c(w,p),∀p∈Hh,c~0(w,p)=c~(w,p),∀p∈Hh.
Therefore, it follows from (17) that
(20)a(w,v)+b(v,q)=0,∀v∈Uh,(qt,p)+c(w,p)+∫0tc~(w,p)dτ=(f,p),∀p∈Hh.
The discontinuous mixed covolume scheme for (2) reads as follows. Seek (wh,qh)∈Uh×Hh such that
(21)a(wh,v)+b(v,qh)=0,∀v∈Uh,(qht,p)+c(wh,p)+∫0tc~(wh,p)dτ=(f,p),∀p∈Hh,
where qh(0)=q~h(0), wh(0)=w~h(0), q~h(0),w~h(0) will be given in Section 4.
Let N>0 be a positive integer; let 0=t0<t1<⋯<tj<⋯<tN=T be a subdivision of time. tj=jΔt(0≤j≤N),Δt=T/N. We use the backward Euler difference quotient
(22)∂tqhj=qhj-qhj-1Δt(j=1,2,…,N)
to approximate the differential quotient ∂qhj/∂t(j=1,2,…,N) and the numerical integration Δt∑k=0j-1∇·(Dkwhk) to approximate the integration ∫0tj∇·(Dwh)dτ(j=1,2,…,N) in the semidiscrete scheme; then we obtain the backward Euler fully discrete discontinuous mixed covolume scheme for the problem (1): find (whj,qhj)∈Uh×Hh(j=1,2,…,N), such that
(23)a(whj,v)+b(v,qhj)=0,∀v∈Uh,(∂tqhj,p)+c(whj,p)+c~(Δt∑k=0j-1whk,p)=(f,p),c~(Δt∑k=0j-1whk,p)=∀p∈Hh,j=1,2,…,N,
where qh0=q~h(0), wh0=w~h(0), q~h(0), w~h(0) will be given in Section 4.
We define the following norms for u∈U(h):
(24)|∥u∥|div2=∥u∥2+∥∇h·u∥2+∑e∈Γ1he∫e[u]2ds,|∥u∥|12=∥u∥2+|u|1,h2+∑e∈Γ1he∫e[u]2ds,|∥u∥|2=|∥u∥|12+∑K∈ThhK2|u|2,K2,
where ∇h·u is the function whose restriction to each element K∈Th is equal to ∇·u and |u|1,h2=∑K∈Th|u|1,K2.
We will introduce some useful Lemmas; for more details, see [6].
Lemma 1.
For u,v∈U(h), we have
(25)a(u,v)≤C|∥u∥|div|∥v∥|div.
Lemma 2.
For (u,p)∈U(h)×L2(Ω), we have
(26)b(u,p)=-c(u,p).
Lemma 3.
For (u,p)∈U(h)×L2(Ω), we have
(27)b(u,p)≤C|∥u∥|(∥p∥+(∑K∈ThhK2|p|1,K2)1/2);
if (u,p)∈Uh×Hh, then
(28)b(u,p)≤C|∥u∥|·∥p∥,c~(u,p)≤C|∥u∥|·∥p∥.
Lemma 4.
Let Zh={u∣u∈Uh,c(u,p)=0,∀p∈Hh}; for any u∈Zh, there is a constant C0 independent of h such that, for α is large enough,
(29)a(u,u)≥C0|∥u∥|div2.
3. Existence and Uniqueness for Discontinuous Mixed Covolume Approximations
In this section, we prove the discontinuous mixed covolume formulation has a unique solution in the finite element space Uh×Hh.
Theorem 5.
The semidiscrete discontinuous mixed covolume scheme (21) has a unique solution in the space Uh×Hh.
Proof.
Only prove that homogenous equation
(30)a(wh,u)+b(u,qh)=0,∀u∈Uh,(qht,p)+c(wh,p)+∫0tc~0(wh,p)=0,∀p∈Hh,qh(0)=0,wh(0)=0,
of (21) exists unique zero solution since the number of unknowns is the same as number of line equations.
By letting v=wh in the first formula of (30) and p=qh in the second formula of (30), using Lemma 2, the sum of (30) gives
(31)a(wh,wh)+(qht,qh)=-∫0tc~0(wh,qh).
Using (1/2)(d/dt)(qh,qh)=(qht,qh) and Lemmas 3 and 4, we have that
(32)C0|∥wh∥|div2+12ddt∥qh∥2≤C∫0t|∥wh∥|∥qh∥dτ≤C∫0t|∥wh∥|div∥qh∥dτ.
Using Hölder inequality and Gronwall Lemma, we get
(33)C0|∥wh∥|div2+12ddt∥qh∥2≤C∫0t∥qh∥2dτ.
Integrating the above formula, we get
(34)2C0∫0t|∥wh∥|div2dτ+∥qh∥2≤C∫0t∫0s∥qh∥2dτds≤C∫0t∥qh∥2dτ.
Then ∥qh∥=0, |∥wh∥|div=0. So qh=0, wh=0, t∈(0,T]. This completes the proof.
Theorem 6.
The fully discrete discontinuous mixed covolume method defined in (23) has a unique solution in the finite element space Uh×Hh if Δt is sufficiently small.
Proof.
Only prove that homogenous equation
(35)a(whj,v)+b(v,qhj)=0,∀v∈Uh,(∂tqhj,p)+c(whj,p)+c~(Δt∑k=0j-1whk,p)=0,xxxxxxxxxxx∀p∈Hh,j=1,2,…,N,
of (23) exists unique zero solution since the number of unknowns is the same as number of line equations.
By letting v=whj in the first formula of (35) and p=qhj in the second formula of (35), using Lemma 2, the sum of (35) gives
(36)a(whj,whj)+(∂tqhj,qhj)=-c~(Δt∑k=0j-1whk,qhj),xxxxxxxxxxxxxxxxxxzxj=1,2,…,N.
Using Lemmas 3 and 4 and
(37)(∂tqhj,qhj)=1Δt(qhj-qhj-1,qhj)=12Δt[(qhj,qhj)-(qhj-1,qhj-1)+(qhj-qhj-1,qhj-qhj-1)]=12Δt[∥qhj∥2-∥qhj-1∥2+∥qhj-qhj-1∥2]≥12Δt[∥qhj∥2-∥qhj-1∥2],
we have from (36) that
(38)2C0Δt|∥whj∥|div2+∥qhj∥2-∥qhj-1∥2≤CΔt[(Δt)2|∥∑k=0j-1whk∥|div2+∥qhj∥2],j=1,2,…,N.
Adding the above inequality with j from 1 to i, using qh0=0 and the discrete Gronwall inequality, when Δt is sufficiently small, we have
(39)2C0Δt∑j=1i|∥whj∥|div2+∥qhi∥2≤0(i=1,2,…,N).
Hence we have ∥qhi∥2=0 and |∥whi∥|div2=0(i=1,2,…,N); that is, qhi=0 and whi=0(i=1,2,…,N). This completes the proof.
4. A Discontinuous Mixed Covolume Elliptic Projection
Define an operator πK from H1(K) to P1(K) by requiring that, for any ∀u∈H1(K),
(40)∫eiπKuds=∫eiuds,(i=1,2,3),
where ei(i=1,2,3) are the three sides of the element K∈Th. It was proved in [5] that
(41)|πKu-u|s,K≤h2-s|u|2,K,∀u∈H2(K),(s=0,1,2).
For any u∈H01(Ω)2, define Π1u∈Uh by
(42)(Π1u)i|K=ΠKui,∀K∈Th,(i=1,2).
Using the definition of Π1 and integration by parts, we can show that
(43)c(u-Π1u,p)=0,∀p∈Hh.
It was proved in [6] that
(44)|∥w-Π1w∥|div≤ch∥w∥2.
Let Π2 be the projection from L02(Ω) to the finite element space Hh.
Define a discontinuous mixed covolume elliptic projection by requiring that, finding w~h,q~n: (0,t)→Uh×Hh, such that
(45)a(w-w~h,v)+b(v,q-q~h)=0,∀v∈Uh,c(w-w~h,p)=0,∀p∈Hh.
It was proved in [15] that the above formula has a unique solution and the error estimates in the following Theorem 7.
Theorem 7.
Let (w~h,q~h)∈Uh×Hh be the solution of (45) and (w,q)∈H2(Ω)2×H1(Ω) the solution of (20). Then there exists a positive constant C independent of h such that
(46)|∥w-w~h∥|div+∥q-q~h∥≤Ch(∥w∥2+∥q∥1).
Theorem 8.
Let (w~h,q~h)∈Uh×Hh be the solution of (45) and (w,q)∈H2(Ω)2×H1(Ω) the solution of (20). Then there exists a positive constant C independent of h such that
(47)|∥(w-w~h)t∥|div+∥(q-q~h)t∥≤Ch(∥wt∥2+∥qt∥1+∥w∥2+∥q∥1).
Differentiating each equation of (45) on t and using (43), (44) we can prove this theorem in the same way as [15].
5. Error Estimates for Semidiscrete Method
In this section, we will establish the error estimates in the H(div) and L2 norms for the semidiscrete discontinuous mixed covolume method.
Theorem 9.
Let (wh,qh)∈Uh×Hh be the solution of (21) and qh(0)=q~h(0), (w,q)∈H2(Ω)2×H1(Ω) the solution of (2). Then there exists a positive constant C independent of h such that
(48)|∥w-wh∥|div+∥q-qh∥≤Ch[∫0t(∥wt∥2+∥qt∥1+∥w∥2+∥q∥1)dτ+∥w∥2+∥q∥1].
Proof.
Let ξ=q~h-qh, η=w~h-wh. Subtracting the two equations of (21) from those of (20), respectively, we have
(49)a(w-wh,v)+b(v,q-qh)=0,∀v∈Uh,(qt-qht,p)+c(w-wh,p)+∫0tc~(w-wh,p)dτ=0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhi∀p∈Hh.
Using (45), we have
(50)a(η,v)+b(v,ξ)=0,∀v∈Uh,(qt-qht,p)+c(η,p)=-∫0tc~(η,p)dτ-∫0tc~(w-w~h,p)dτ,∀p∈Hh.
Differentiating the first equation of (50) on t, we have that
(51)a(ηt,v)+b(v,ξt)=0,∀v∈Uh.
By letting p=ξt in the second formula of (50) and letting v=η in (51), using Lemma 2, the sum of them gives
(52)a(ηt,η)+(ξt,ξt)=((p~h-p)t,ξt)-∫0tc~(η,ξt)dτ-∫0tc~(w-w~h,ξt)dτ.
Using
(53)a(ηt,η)=12ddta(η,η)-12(Mtη,η)
and Lemmas 1 and 3 gives
(54)12ddta(η,η)+∥ξt∥2hhhhh≤C[∫0t|∥η∥|div∥ξt∥dτ∥ξt∥·∥(p~h-p)t∥+|∥η∥|div2hhhhhhhhhh+∫0t|∥η∥|div∥ξt∥dτ+∫0t|∥w-w~h∥|div∥ξt∥dτ].
Multiplying the equation above with 2, integrating them from 0 to t and using Hölder inequality, ϵ-inequality, Gronwall inequality, Lemma 4, and (47), we can get
(55)|∥η∥|div2+1C0(∫0t∥ξt∥dτ)2≤Ch2(∫0t(∥wt∥2+∥qt∥1+∥w∥2+∥q∥1)dτ)2,
so
(56)|∥η∥|div≤Ch∫0t(∥wt∥2+∥qt∥1+∥w∥2+∥q∥1)dτ,(57)∫0t∥ξt∥dτ≤Ch∫0t(∥wt∥2+∥qt∥1+∥w∥2+∥q∥1)dτ;
hence
(58)∥ξ∥≤∫0t∥ξt∥dτ≤Ch∫0t(∥wt∥2+∥qt∥1+∥w∥2+∥q∥1)dτ.
Now, using the triangle inequality, (46), (56), and (58), we get
(59)|∥w-wh∥|div+∥q-qh∥≤Ch[∫0t(∥wt∥2+∥qt∥1+∥w∥2+∥q∥1)dτ+∥w∥2+∥q∥1].
The proof is complete.
6. Error Estimates for Fully Discrete Method
Let ξj=q~hj-qhj, ζj=q~hj-qj, ηj=w~hj-whj(j=0,1,…,N), and then the error estimates for the backward Euler fully discrete discontinuous mixed covolume method in the H(div) and L2 norms are provided in next two theorems.
Theorem 10.
Let (w,q)∈H2(Ω)2×H1(Ω) be the solution of (2) and (whj,qhj)∈Uh×Hh(j=1,2,…,N) the solution of (23) with t=tj(j=1,2,…,N), respectively. If qh0=q~h(0)=q0, wh0=w~h(0)=w0, then there exists a positive constant C independent of h and Δt such that
(60)max0≤i≤N∥qi-qhi∥+max0≤i≤N|∥wi-whi∥|div≤CΔt(∥qtt∥L∞(L2)+∥wt∥L∞(H2))+Ch(∥wt∥L∞(H2)+∥qt∥L∞(H1)∥wt∥L∞(H2)+∥w∥L∞(H2)+∥q∥L∞(H1)).
Proof.
Subtracting the two equations of (23) from (20), respectively, with t=tj(j=0,1,…,N), we can get the error equation:
(61)a(wj-whj,v)+b(v,qj-qhj)=0,∀v∈Uh,(qtj-∂tqhj,p)+c(wj-whj,p)+∫0tjc~(w,p)dτ-c~(Δt∑k=0j-1whk,p)=0,∀p∈Hh,j=1,2,…,N.
Choosing v=ηj(j=1,2,…,N) and p=ξj(j=1,2,…,N) in the two equations of (61), adding them together, and using Lemma 2, discontinuous mixed covolume elliptic projection with t=tj(j=0,1,…,N), we have
(62)a(ηj,ηj)+(∂tξj,ξj)=(∂tqj-qtj,ξj)+(∂tζj,ξj)-∫0tjc~(w-w~h,ξj)dτ-c~(Δt∑k=0j-1ηk,ξj)-∫0tjc~(w~h,ξj)dτ+c~(Δt∑k=0j-1w~hk,ξj).
First, we estimate the left item of (62). Using Lemma 4, we have
(63)a(ηj,ηj)≥C0|∥ηj∥|div2,(∂tξj,ξj)=1Δt(ξj-ξj-1,ξj)=12Δt[(ξj,ξj)-(ξj-1,ξj-1)+(ξj-ξj-1,ξj-ξj-1)]=12Δt[∥ξj∥2-∥ξj-1∥2+∥ξj-ξj-1∥2]>12Δt[∥ξj∥2-∥ξj-1∥2].
Then, we estimate the right item of (62). From
(64)∥∂tqj-qtj∥2=∥1Δt∫tj-1tj(tj-1-t)qttdt∥2≤∫Ω(1Δt∫tj-1tj(tj-1-t)qttdt)2dx≤1(Δt)2∫Ω(∫tj-1tj(tj-1-t)2dt∫tj-1tjqtt2dt)dx≤C(Δt)2∥qtt∥L∞(L2)2,
we have
(65)(∂tqj-qtj,ξj)≤C(Δt)∥qtt∥L∞(L2)∥ξj∥,∥∂tζj∥=∥ζj-ζj-1Δt∥=∥1Δt∫tj-1tjζtdt∥≤1Δt∫tj-1tj∥ζt∥dt≤ChΔt∫tj-1tj(∥wt∥2+∥qt∥1+∥w∥2+∥q∥1)dt≤Ch(∥wt∥L∞(H2)+∥qt∥L∞(H1)+∥w∥L∞(H2)+∥q∥L∞(H1)),
and therefore
(66)(∂tζj,ξj)hhh≤Ch(∥wt∥L∞(H2)+∥qt∥L∞(H1)+∥w∥L∞(H2)+∥q∥L∞(H1))hhhhhi×∥ξj∥.
Using Lemma 3, we can get
(6)-∫0tjc~(w-w~h,ξj)dτ≤C∫0tj|∥w-w~h∥|divdτ∥ξj∥≤Ch(∥w∥L∞(H2)+∥p∥L∞(H1))∥ξj∥,-c~(Δt∑k=0j-1ηk,ξj)≤C(Δt)∑k=0j-1|∥ηk∥|div∥ξj∥,-∫0tjc~(w~h,ξj)dτ+c~(Δt∑k=0j-1w~hk,ξj)=c~(∑k=0j-1∫tktk+1(w~hk-w~h(τ))dτ,ξj)≤C∥∑k=0j-1∫tktk+1∫tkτw~htdsdτ∥∥ξj∥≤C(Δt)∥wt∥L∞(H2)∥ξj∥.
Substituting the estimations above into (62), multiplying them with Δt, adding them with j from 1 to i, and using ξ0=0, we have
(68)∥ξi∥2+2C0Δt∑j=1i|∥ηj∥|div2≤C(Δt)2(∥qtt∥L∞(L2)+∥wt∥L∞(H2))2+C(Δt)2∑k=0j-1|∥ηk∥|div2+CΔt∑j=1i∥ξj∥2+Ch2(∥wt∥L∞(H2)+∥qt∥L∞(H1)+∥w∥L∞(H2)+∥q∥L∞(H1))2.
Using the discrete Gronwall inequality, we have
(69)max0≤i≤N∥ξi∥+max0≤i≤N|∥ηi∥|div≤CΔt(∥qtt∥L∞(L2)+∥wt∥L∞(H2))+Ch(∥wt∥L∞(H2)+∥qt∥L∞(H1)+∥w∥L∞(H2)+∥q∥L∞(H1)).
From the formula above and (46) and using the triangle inequality, we have
(70)max0≤i≤N∥qi-qhi∥+max0≤i≤N|∥wi-whi∥|div≤CΔt(∥qtt∥L∞(L2)+∥wt∥L∞(H2))+Ch(∥wt∥L∞(H2)+∥qt∥L∞(H1)+∥w∥L∞(H2)+∥q∥L∞(H1)).
This completes the proof.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author acknowledges a project supported by the fund of National Natural Science (11171193) and a project of Shandong Province Science and Technology Development Program (2012GGB01198).
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