# Expanded Mixed Finite Element Method for the Two-Dimensional Sobolev Equation

Authors: Qing-li Zhao; Zong-cheng Li; You-zheng Ding
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Expanded mixed finite element method is introduced to approximate the two-dimensional Sobolev equation. This formulation expands the standard mixed formulation in the sense that three unknown variables are explicitly treated. Existence and uniqueness of the numerical solution are demonstrated. Optimal order error estimates for both the scalar and two vector functions are established.


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## Body

## 1. Introduction

In this paper, we consider the following Sobolev equation:(1)ut(x,t)-∇·{a(x,t)∇ut+b1(x,t)∇u(x,t)}=f(x,t),(x,t)∈Ω×(0,T],u(x,t)=0,(x,t)∈∂Ω×[0,T],u(x,0)=u0(x),x∈Ω,
in a bounded domain Ω⊂R2 with Lipschitz continuous boundary ∂Ω. Equation (1) has a wide range of applications in many mathematical and physical problems [1, 2], for example, the percolation theory when the fluid flows through the cracks, the transfer problem of the moisture in the soil, and the heat conduction problem in different materials. So there exists great and actual significance to research Sobolev equation. Up till now, there are some different schemes studied to solve this kind of equation (see [1–4] for instance).The mixed finite element method, which is a finite element method [5] with constrained conditions, plays an important role in the research of the numerical solution for partial differential equations. Its general theory was proposed by Babuska [6] and Brezzi [7]. Falk and Osborn [8] improved their theory and expanded the adaptability of the mixed finite element method. The mixed finite element method (see [9, 10] for instance) is wildly used for the modeling of fluid flow and transport, as it provides accurate and locally mass conservative velocities.The main motivation of the expanded mixed finite element method [11–15] is to introduce three (or more) auxiliary variables for practical problems, while the traditional finite element method and mixed finite element method can only approximate one and two variables, respectively. The expanded mixed finite element method also has some other advantages except introducing more variables. It can treat individual boundary conditions. Also, this method is suitable for differential equation with small coefficient (close to zero) which does not need to be inverted. Consequently, this method works for the problems with small diffusion or low permeability terms in fluid problems. Using this method, we can get optimal order error estimates for certain nonlinear problems, while standard mixed formulation sometimes gives only suboptimal error estimates.The object of this paper is to present the expanded mixed finite element method for the Sobolev equation. We conduct theoretical analysis to study the existence and uniqueness and obtain optimal order error estimates. The rest of this paper is organized as follows. In Section2, the mixed weak formulation and its mixed element approximation are considered. In Section 3, we prove the existence and uniqueness of approximation form. In Section 4, some lemmas are given. In Section 5, optimal order semidiscrete error estimates are established.Throughout the paper, we will useC, with or without subscript, to denote a generic positive constant which does not depend on the discretization parameter h. Vectors will be expressed in boldface. At the same time, we show a useful ε-Cauchy inequality
(2)ab≤εa2+14εb2,ε>0.


## 2. Mixed Weak Form and Mixed Element Approximation

For1≤q≤+∞ and m any nonnegative integer, let
(3)Wm,q(Ω)={f∈Lq(Ω)∣Dαf∈Lq(Ω),|α|≤m}
denote the Sobolev spaces [16] endowed with the norm
(4)∥f∥m,q,Ω=(∑|α|≤m‍∥Dαf∥Lq(Ω)q)1/q
(the subscript Ω will always be omitted).LetHm(Ω)=Wm,2(Ω) with the norm ∥·∥m=∥·∥m,2. The notation ∥·∥ will mean ∥·∥L2(Ω) or ∥·∥L2(Ω)2. We denote by (,) the inner product in either L2(Ω) or L2(Ω)2; that is,
(5)(ξ,η)=∫Ω‍ξηdx,(ξ,η)=∫Ω‍ξ·ηdx.
The notation 〈,〉 denotes the L2-inner product on the boundary of Ω(6)〈ξ,η〉=∫∂Ω‍ξηds.To formulate the weak form, letb=b1/a; we introduce two vector variables
(7)λ=-∇ut-b∇u,σ=aλ=-a∇ut-b1∇u.
Letting c=-∇b, we rewrite (1) as the following system:
(8)ut+∇·σ=f,(x,t)∈Ω×(0,T],λ+∇ut-∇(bu)+cu=0,(x,t)∈Ω×(0,T],σ-aλ=0,(x,t)∈Ω×(0,T],u(x,t)=0,(x,t)∈∂Ω×(0,T],u(x,0)=u0(x),x∈Ω.
Let n be the unit exterior normal vector to the boundary of Ω. Then (8) is formulated in the following expanded mixed weak form: find (u,λ,σ)∈W×Λ×V, such that
(9)(ut,w)+(divσ,w)=(f,w),∀w∈W,0<t≤T,(λ,v)-(ut+bu,divv)+(cu,v)=0,∀v∈V,0<t≤T,(σ,τ)-(aλ,τ)=0,∀τ∈Λ,0<t≤T,(u(0),w)=(u0,w),∀w∈W,
where
(10)W=L2(Ω),Λ=L2(Ω)2,V=H(div,Ω)={v∈L2(Ω)2∣∇·v∈L2(Ω)}.LetTh be a quasiregular polygonalization of Ω (by triangles or rectangles), with h being the maximum diameter of the elements of the polygonalization. Let Wh×Λh×Vh be a conforming mixed element space with index k and discretization parameter h. Wh×Λh×Vh is an approximation to W×Λ×V. There are many conforming (or compatible) mixed element function spaces such as Raviart-Thomas elements [17], BDFM elements [18, 19]. Some RT type mixed elements are listed in Table 1. Here, Pk(T) is the polynomial up to k order in two-dimensional domain used in triangle, while Qk,l(T) is the polynomial up to k,l in each dimension used in rectangle.Table 1
Some RT type mixed elements.
Dim
Element
V
h
(
E
)
W
h
(
E
)
2D
Triangle
R
T
k
(
T
)
=
P
k
(
E
)
2
⊕
x
P
k
(
E
)
P
k
(
E
)
2D
Rectangle
R
T
[
k
]
(
T
)
=
Q
k
+
1
,
k
(
E
)
⊕
Q
k
,
k
+
1
(
E
)
Q
k
,
k
(
E
)Replacing the three variables by their approximation, we get the expanded mixed finite element approximation problem: find(uh,λh,σh)∈Wh×Λh×Vh, such that
(11)(uh,t,w)+(divσh,w)=(f,w),∀w∈Wh,0<t≤T,(λh,v)-(uh,t+buh,divv)+(cuh,v)=0,∀v∈Vh,0<t≤T,(σh,τ)-(aλh,τ)=0,∀τ∈Λh,0<t≤T,(u(0),w)=(u~0,w),∀w∈Wh,
where
(12)Wh(E)={ω∈W:ω|E∈Wh(E),∀E∈Th},Λh(E)={τ∈Λ:τ|E∈Vh(E),∀E∈Th},Vh(E)={v∈V:v|E∈Vh(E),∀E∈Th}.The error analysis next makes use of three projection operators. The first operator is the Raviart-Thomas projection (or Brezzi-Douglas-Marini projection)Πh:V→Vh; Πh satisfies
(13)(∇·(v-Πhv),ωh)=0,∀ωh∈Wh.
The following approximation properties are well known.
(14)∥v-Πhv∥≤Chr∥v∥r,12≤r≤k+1,(15)∥∇·(v-Πhv)∥r≤Chr∥∇·v∥r,0≤r≤k+1.
The other two operators are the standard L2-projection [5] Ph and Rh onto Wh and Λh, respectively,
(16)(ω-Phω,∇·vh)=0,∀ω∈W,vh∈Vh,(τ-Rhτ,τh)=0,∀τ∈Λ,τh∈Λh.
They have the approximation properties
(17)∥ω-Phω∥≤Chr∥ω∥r,0≤r≤k+1,∥τ-Rhτ∥≤Chr∥μ∥r,0≤r≤k+1.
The two projections Πh and Ph preserve the commuting property
(18)div∘Πh=Ph∘div:H1(Ω)2⟶Wh.


## 3. Existence and Uniqueness of Approximation Form

In this section, we consider the existence and uniqueness of the solution of (11).Lemma 1.
Equation (11) has a unique solution.Proof.
In fact, this equation is linear; it suffices to show that the associated homogeneous system(19)(uh,t,w)+(divσh,w)=0,∀w∈Wh,0<t≤T,(λh,v)-(uh,t+buh,divv)+(cuh,v)=0,∀v∈Vh,0<t≤T,(σh,τ)-(aλh,τ)=0,∀τ∈Λh,0<t≤T,(u(0),w)=0,∀w∈Wh,
has only the trivial solution. In the first equation of (19), if we take ω=uh,t and ω=divσh, respectively, then we have that
(20)∥uh,t∥≤∥divσh∥,∥divσh∥≤∥uh,t∥.
By (20), it is easy to see that
(21)∥divσh∥=∥uh,t∥.
Choosing ω=divσh, v=σh, and τ=λh in (19), we get
(22)(aλh,λh)+(divσh,divσh)=(buh,divσh)-(cuh,σh).
In the third equation of (19), letting τ=σh and τ=λh, respectively, then
(23)∥σh∥≤C∥λh∥,(24)∥λh∥≤C∥σh∥.
Using the ε-Cauchy inequality to (22), we have
(25)a0∥λh∥2+∥divσh∥2≤C∥uh∥2+12∥divσh∥2+a02∥λh∥2.
Note that (21), we deserve
(26)∥λh∥≤C∥uh∥,(27)∥uh,t∥≤C∥uh∥.
From (27), we know
(28)∥uh∥=∥∫0t‍uh,tdt∥≤C∫0t‍∥uh,t∥dt≤C∫0t‍∥uh∥dt.
Using Gronwall’s inequality to (28), we can prove uh=0. Further, from (26) and (23), we know that λh=0, σh=0. The proof is completed.


## 4. Some Lemmas

In the study of parabolic equations, we usually introduce a mixed elliptic projection associated with our equations. Define a map: find(u~h,λ~h,σ~h)∈Wh×Λh×Vh, such that
(29)(div(σ-σ~h),w)=0,∀w∈Wh,0<t≤T,(λ-λ~h,v)-(ut-u~h,t+b(u-u~h),divv)+(c(u-u~h),v)=0,∀v∈Vh,0<t≤T,(σ-σ~h,τ)-(a(λ-λ~h),τ)=0,∀τ∈Λh,0<t≤T,(u(0)-u~h(0),w)=0,∀w∈Wh.
Similarly to Lemma 1, we can prove that system (29) has a unique solution.Now we give some error estimates of(u~h,λ~h,σ~h). Define
(30)u-u~h=(phu-u~h)+(u-phu)=u1+u2,λ1=λ-λ~h,σ1=σ-σ~h.
System (29) can be rewritten as follows:
(31)(divσ1,w)=0,∀w∈Wh,(λ1,v)-(u1,t+b(u1+u2),divv)+(c(u1+u2),v)=0,∀v∈Vh,(σ1,τ)-(aλ1,τ)=0,∀τ∈Λh.(u(0)-u~h(0),w)=0,∀w∈Wh.
Now we consider the estimates of u1 and u1,t.Lemma 2.
Let(u1,λ1,σ1) be the solution of system (31). If u, λ, and σ are sufficiently smooth, then there exist positive constants C such that
(32)∥u1∥≤C∫0t‍(h∥λ1∥+h1-δk0∥λ1∥+h∥u2∥+∥u2∥-1)dt,∥u1,t∥≤C(h∥λ1∥+h1-δk0∥λ1∥+h∥u2∥+∥u2∥-1+∥u1∥),
where δk0=0 for k=0, and δk0=0 for k≥1.Proof.
Letϕ∈H2(Ω)⋂‍H01(Ω) be the solution of the following problem:
(33)∇·(a∇ϕ)=ψ,∀x∈Ω,ϕ|∂Ω=0.
Then we know
(34)∥ϕ∥2≤C∥ψ∥.
For 0<t≤T, from the second equation of (31), we have that
(35)(u1,t,ψ)=(u1,t,div(a∇ϕ))(u1,t,ψ)=(u1,t,divΠh(a∇ϕ))(u1,t,ψ)=(λ1,Πh(a∇ϕ))+(c(u1+u2),Πh(a∇ϕ))(u1,t,ψ)=-(b(u1+u2),divΠh(a∇ϕ))(u1,t,ψ)=I1+I2+I3.
It is easy to see that
(36)I1=(λ1,Πh(a∇ϕ))I1=(λ1,Πh(a∇ϕ)-a∇ϕ)+(λ1,a∇ϕ)I1=(λ1,Πh(a∇ϕ)-a∇ϕ)+(aλ1,∇ϕ)I1=E1+E2,E1≤Ch∥λ1∥∥ϕ∥2≤Ch∥λ1∥∥ψ∥,E2=(aλ1,∇(ϕ-phϕ))+(aλ1,∇(phϕ))E2=(aλ1,∇(ϕ-phϕ))+(σ1,∇(phϕ))E2=(aλ1,∇(ϕ-phϕ))-(divσ1,(phϕ))E2=(aλ1,∇(ϕ-phϕ))E2≤C∥λ1∥∥∇(ϕ-phϕ)∥E2≤Ch1-δko∥ϕ∥2∥λ1∥E2≤Ch1-δko∥ψ∥∥λ1∥.
Now, we estimate I2,
(37)(cu1,Πh(a∇ϕ))≤C∥u1∥∥Πh(a∇ϕ)-a∇ϕ+a∇ϕ∥≤C∥u1∥(h∥∇ϕ∥+∥∇ϕ∥)≤C∥u1∥∥ψ∥,(cu2,Πh(a∇ϕ))=(cu2,Πh(a∇ϕ)-a∇ϕ)+(cu2,a∇ϕ)≤C(h∥u2∥+∥u2∥-1)∥ψ∥.
We turn to consider I3,
(38)(-bu1,divΠh(a∇ϕ))≤C∥u1∥∥∇ϕ∥≤C∥u1∥∥ψ∥,(-bu2,divΠh(a∇ϕ))≤-((b-ph0b)u2,divΠh(a∇ϕ))-∑e∈Th‍((ph0b)u2,divΠh(a∇ϕ))e=-((b-ph0b)u2,divΠh(a∇ϕ))≤Ch∥u2∥∥ψ∥,
where ph0b is the piecewise constant interpolation of function b. Using the previous estimates, we obtain
(39)∥u1,t∥≤C(h∥λ1∥+h1-δk0∥λ1∥+h∥u2∥+∥u2∥-1+∥u1∥).
So we deserve
(40)∥u1∥=∥∫0t‍u1,tdt∥∥u1∥≤C∫0t‍∥u1,t∥dt∥u1∥≤C∫0t‍(∥u2∥-1h∥λ1∥+h1-δk0∥λ1∥+h∥u2∥+∥u2∥-1+∥u1∥h∥λ1∥+h1-δk0∥λ1∥+h∥u2∥)dt.
Applying Gronwall’s inequality to (40), we have
(41)∥u1∥≤C∫0t‍(h∥λ1∥+h1-δk0∥λ1∥+h∥u2∥+∥u2∥-1)dt.
Noting the estimates of I1, I2, and I3, the proof is completed.Define(42)∥u,λ,σ∥r=∥u∥r+∥λ∥r+∥σ∥r,|∥u,λ,σ∥|r=∫0t‍(∥u∥r+∥λ∥r+∥σ∥r)dt.Lemma 3.
Let(u,λ,σ) and (u~h,λ~h,σ~h) be the solution of (19) and (29), respectively. If u, λ, and σ are sufficiently smooth, then there exist positive constants C such that(I)
ifk≥0, then
(43)∥div(σ-σ~h)∥≤Chr∥σ∥r+1,0≤r≤k+1,(44)∥phu-u~h∥≤Chr+1-δk0|∥u,λ,σ∥|r,1≤r≤k+1,(45)∥phut-u~h,t∥≤Chr+1-δk0∥u,λ,σ∥r,1≤r≤k+1,(II)
ifk=0, then
(46)∥u-u~h∥≤Ch(∥u∥1+|∥u,λ,σ∥|1),(47)∥ut-u~h,t∥≤Ch(∥ut∥1+∥u,λ,σ∥1+|∥u,λ,σ∥|1),(48)∥λ-λ~h∥≤Ch(∥u,λ,σ∥1+|∥u,λ,σ∥|1),(49)∥σ-σ~h∥≤Ch(∥u,λ,σ∥1+|∥u,λ,σ∥|1),(III)
ifk≥1,2≤r≤k+1, then
(50)∥u-u~h∥≤Chr(∥u∥r+|∥u,λ,σ∥|r-1),(51)∥ut-u~h,t∥≤Chr(∥ut∥r+∥u,λ,σ∥r+|∥u,λ,σ∥|r-1),(52)∥λ-λ~h∥≤Chr(∥u,λ,σ∥r+|∥u,λ,σ∥|r-1),(53)∥σ-σ~h∥≤Chr(∥u,λ,σ∥r+|∥u,λ,σ∥|r-1).Proof.
For0<t≤T, the proof proceeds in three steps as follows.
(I) In the first equation of (29), taking ω=divσ1, we know
(54)(divσ1,divσ1)=(divσ1,div(σ-Πhσ))≤∥divσ1∥∥div(σ-Πhσ)∥,
which implies that
(55)∥divσ1∥≤∥div(σ-Πhσ)∥∥divσ1∥≤Chr∥σ∥r+1,0≤r≤k+1.
Choosing ω∈Wh, note that (div(σ-σ~h),ω)=0 and (div(σ-Πhσ),ω)=0, we get
(56)div(Πhσ-σ~h)=0.
Combing (15) with (56), we can prove (43).
(II) In (29), letting v=Πhσ-σ~h, together with (43), we obtain
(57)(λ-λ~h,Πhσ-σ~h)-(c(u-u~h),Πhσ-σ~h)=0.
Hence
(58)(Rhλ-λ~h,Πhσ-σ~h)-(c(u-u~h),Πhσ-σ~h)=0.
In the third equation of (29), choosing τ=Rhλ-λ~h, we get
(59)(σ-σ~h,Rhλ-λ~h)=(a(λ-λ~h),Rhλ-λ~h).
So we know
(60)(σ-Πhσ,Rhλ-λ~h)+(Πhσ-σ~h,Rhλ-λ~h)=(a(λ-λ~h),Rhλ-λ~h).
Note that (44) and (45), we obtain
(61)(Πhσ-σ~h,Rhλ-λ~h)=(c(uh-u~h),Πhσ-σ~h)=(a(λ-λ~h),Rhλ-λ~h)-(σ-Πhσ,Rhλ-λ~h),
which implies that
(62)(Πhσ-σ~h,Rhλ-λ~h)≤C∥u-u~h∥∥Πhσ-σ~h∥≤C(∥u1∥+∥u2∥)∥Πhσ-σ~h∥.
Now we estimate ∥Πhσ-σ~h∥,
(63)(σ-σ~h,Πhσ-σ~h)=(a(λ-λ~h),Πhσ-σ~h).
It is easy to see that
(64)∥Πhσ-σ~h∥2=(Πhσ-σ,Πhσ-σ~h)+(a(λ-λ~h),Πhσ-σ~h)=(Πhσ-σ,Πhσ-σ~h)+(a(λ-Rhλ),Πhσ-σ~h)+(a(Rhλ-λ~h),Πhσ-σ~h)≤C(∥Πhσ-σ∥+∥λ-Rhλ∥+∥u1∥+∥u2∥)×∥Πhσ-σ~h∥.
By the properties of projection Πh, we get
(65)∥Πhσ-σ~h∥≤Chr(∥u∥r+∥λ∥r+∥σ∥r+∥u1∥),0≤r≤k+1,
which proves (49) and (53).
Now we estimate∥Rhλ-λ~h∥,
(66)(a(Rhλ-λ~h),Rhλ-λ~h)=(a(λ-λ~h,Rhλ-λ~h)+(a(Rhλ-λ),Rhλ-λ~h)=(σ-Πhσ,Rhλ-λ~h)+(Πhσ-σ~h,Rhλ-λ~h)+(a(Rhλ-λ),Rhλ-λ~h)=(σ-Πhσ,Rhλ-λ~h)+(c(u-u~h),Πhσ-σ~h)+(a(Rhλ-λ),Rhλ-λ~h)≤(∥σ-Πhσ∥+C∥λ-Rhλ∥)∥Rhλ-λ~h∥+C∥u1+u2∥∥Πhσ-σ~h∥≤C(∥σ-Πhσ∥2+∥u1∥2+∥u2∥2+∥Rhλ-λ∥2)+c02∥Rhλ-λ~h∥2,
where
(67)(a(Rhλ-λ~h),Rhλ-λ~h)≥c0∥Rhλ-λ~h∥2.
From (66) and (67), we know
(68)∥Rhλ-λ~h∥≤C(hr(∥u∥r+∥λ∥r+∥σ∥r)+∥u1∥),∥λ-λ~h∥≤C(hr(∥u∥r+∥λ∥r+∥σ∥r)+∥u1∥),0≤r≤k+1.
By Lemma 2, if k=0, we have that
(69)∥λ-λ~h∥=∥λ1∥∥λ-λ~h∥≤Ch(∫0t‍∥u∥1+∥λ∥1+∥σ∥1∥λ-λ~h∥=+∫0t‍(∥u∥1+∥λ∥1+∥σ∥1)dτ).
If k≥1,2≤r≤k+1, we have that
(70)∥λ-λ~h∥≤Chr(∫0t‍∥u∥r+∥λ∥r+∥σ∥r+∫0t‍(∥u∥r-1+∥λ∥r-1∥σ∥r-1)dt).
Using the estimate of ∥Πhσ-σ~h∥, we obtain
(71)∥σ-σ~h∥≤C(hr(∥u∥r+∥λ∥r+∥σ∥r)+∥u1∥),0≤r≤k+1.
If k=0, we get
(72)∥σ-σ~h∥≤Ch(∫0t‍∥u∥1+∥λ∥1+∥σ∥1+∫0t‍(∥u∥1+∥λ∥1+∥σ∥1)dt).
If k≥1,2≤r≤k+1, we get
(73)∥σ-σ~h∥≤Chr(∫0t‍∥u∥r+∥λ∥r+∥σ∥r+∫0t‍(∥u∥r-1+∥λ∥r-1+∥σ∥r-1)dt).
(III) By Lemma2, we have that
(74)∥phu-u~h∥=∥u1∥≤Chr+1-δk0∥phu-u~h∥=×∫0t‍(∥u∥r+∥λ∥r+∥σ∥r)dτ,0≤r≤k+1,∥phut-u~h,t∥≤Chr+1-δk0(∫0t‍∥u∥r+∥λ∥r+∥σ∥r+∫0t‍(∥u∥r+∥λ∥r+∥σ∥r)dt),0≤r≤k+1.
If k=0, we know
(75)∥u-u~h∥≤∥u-phu∥+∥u1∥∥u-u~h∥≤Ch(∥u∥1+∫0t‍(∥u∥1+∥λ∥1+∥σ∥1)dt),∥ut-u~h,t∥≤Ch(∫0t‍∥ut∥1+∥u∥1+∥λ∥1+∥σ∥1+∫0t‍(∥u∥1+∥λ∥1+∥σ∥1)dt).
If k≥1,2≤r≤k+1, we know
(76)∥u-u~h∥≤Chr(∥u∥r+∫0t‍(∥u∥r-1+∥λ∥r-1+∥σ∥r-1)dt),∥ut-u~h,t∥≤Chr(∫0t‍∥ut∥r+∥u∥r-1+∥λ∥r-1+∫0t‍(∥u∥r-1+∥λ∥r-1+∥σ∥r-1)dt).
The proof is completed.Remark 4.
The estimate results of (46)–(53) are optimal. If k≥1, the estimate results of (43)–(45) are superconvergent.


## 5. Main Result

In this section, we consider error estimates for the continuous-in-time mixed finite element approximation. Define(77)u-uh=(u-u~h)+(u~h-uh)=u3+u4,λ-λh=(λ-λ~h)+(λ~h-λh)=λ1+λ2,σ-σh=(σ-σ~h)+(σ~h-σh)=σ1+σ2,∥u,ut,λ,σ∥r=∥u∥r+∥ut∥r+∥λ∥r+∥σ∥r,|∥u,ut,λ,σ∥|r=∫0t‍(∥u∥r+∥ut∥r+∥λ∥r+∥σ∥r)dt.Theorem 5.
Let(u,λ,σ) and (uh,λh,σh) be the solution of (9) and (11), respectively. If u, λ, and σ are sufficiently smooth, then there exist positive constants C such that(I)
ifk=0, then
(78)∥u-uh∥≤Ch(∥u∥1+|∥u,ut,λ,σ∥|1),(79)∥ut-uh,t∥≤Ch(∥u,ut,λ,σ∥1+|∥u,λ,σ∥|1),(80)∥λ-λh∥≤Ch(∥u,ut,λ,σ∥1+|∥u,ut,λ,σ∥|1),(81)∥σ-σh∥≤Ch(∥u,ut,λ,σ∥1+|∥u,ut,λ,σ∥|1),(82)∥div(σ-σh)∥≤Ch(∥u∥1+∥ut∥1+∥σ∥2+|∥u,ut,λ,σ∥|1),(II)
ifk≥1 and 2≤r≤k+1, then
(83)∥u-uh∥≤Chr(∥u∥r+|∥u,ut,λ,σ∥|r-1),∥ut-uh,t∥≤Chr(∫0t‍∥ut∥r+∥u∥r-1+∥λ∥r-1+∥σ∥r-1+∫0t‍(∥u∥r-1+∥ut∥r+∥λ∥r-1+∥σ∥r-1)dt),∥λ-λh∥≤Chr(∥u,λ,σ∥r∫0t‍+∫0t‍(∥ut∥r+∥u∥r-1+∥λ∥r-1+∥σ∥r-1)dt),∥σ-σh∥≤Chr(∫0t‍∥u,λ,σ∥r+∫0t‍(∥ut∥r+∥u∥r-1+∥λ∥r-1+∥σ∥r-1)dt),(84)∥div(σ-σh)∥≤Chr(∫0t‍∥ut∥r+∥λ∥r+∥σ∥r+1+∥u∥r-1+∫0t‍(∥ut∥r+∥u∥r-1+∥λ∥r-1+∥σ∥r-1)dt).Proof.
From (9) and (19), we have the following error equation:
(85)(u4,t+u3,t,w)+(divσ2,w)=0,∀w∈Wh,0<t≤T,(σ2,τ)-(aλ2,τ)=0,∀τ∈Λh,0<t≤T,(λ2,v)-(u4,t+bu4,divv)+(cu4,v)=0,∀v∈Vh,0<t≤T,u4(0)=0.
In (85), choosing ω=u4,t+ph(bu4), τ=λ2, and v=σ2, we have
(86)(λ2,σ2)+(u4,t,u4,t)=-(u3,t,u4,t+ph(bu4))-(u4,t,ph(bu4))-(cu4,σ2).
In the second equation of (85), letting τ=σ2, we know
(87)σ2≤Cλ2.
Further, using ε-inequality to (86), we get
(88)∥u4,t∥2+∥λ2∥2≤C(∥u4∥2+∥u3,t∥2).
Hence,
(89)∥u4∥=∥∫0t‍u4,tdτ∥∥u4∥≤C∫0t‍∥u4,t∥dτ≤C∫0t‍(∥u4∥+∥u3,t∥)dt.
Using Gronwall’s inequality to (89), we obtain
(90)∥u4∥≤C∫0t‍∥u3,t∥dt.
Further, we know
(91)∥σ2∥+∥u4,t∥+∥λ2∥≤C(∥u3,t∥+∫0t‍∥u3,t∥dt).
Noting the first equation of (78), we get
(92)∥divσ2∥≤∥u3,t∥+∥u4,t∥≤C(∥u3,t∥+∫0t‍∥u3,t∥dt).
So we have that
(93)∥u-uh∥≤∥u3∥+C∫0t‍∥u3,t∥dt,∥ut-uh,t∥≤C(∥u3,t∥+∫0t‍∥u3,t∥dt),∥λ-λh∥≤∥λ1∥+C(∥u3,t∥+∫0t‍∥u3,t∥dt),∥σ-σh∥≤∥σ1∥+C(∥u3,t∥+∫0t‍∥u3,t∥dt),∥div(σ-σh)∥≤∥divσ1∥+∥divσ2∥.
Together with the results of Lemmas 2 and 3, the proof is completed.Remark 6.
The estimate results of (78)–(84) are optimal.



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*Source: 934973-2013-06-24.xml*