# A Time Discontinuous Galerkin Finite Element Method for Quasi-Linear Sobolev Equations

Authors: Hong Yu; Tongjun Sun
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

We present a time discontinuous Galerkin finite element scheme for quasi-linear Sobolev equations. The approximate solution is sought as a piecewise polynomial of degree in time variable at mostq
-
1 with coefficients in finite element space. This piecewise polynomial is not necessarily continuous at the nodes of the partition for the time interval. The existence and uniqueness of the approximate solution are proved by use of Brouwer’s fixed point theorem. An optimal L
∞
(
0
,
T
;
H
1
(
Ω
)
)-norm error estimate is derived. Just because of a damping term u
x
x
t included in quasi-linear Sobolev equations, which is the distinct character different from parabolic equation, more attentions are paid to this term in the study. This is the significance of this paper.


---

## Body

## 1. Introduction

The time discontinuous Galerkin method was proposed by Delfour in 1981 [1] to analyze the ordinary differential equations. As well known, the fully discrete scheme for unsteady partial differential equations is normally derived by first discretizing in the space variables by the Galerkin finite element method and then replacing the resulting system of ordinary differential equation with respect to time by the finite difference method in the time variable. In [2], the time discontinuous Galerkin method was presented to approximate the heat conduction equation. By applying the Galerkin method to time variable, the purpose of this method was to find the approximate solution as a piecewise polynomial of degree in time variable t at most q
-
1 with coefficients in finite element space. The piecewise polynomial was not necessarily continuous at the nodes of the partition for the time interval. So, a similar processing method can be used to analyze the space and time variables in the definition and analysis. Unifying the space and time variables, time discontinuous finite element method overcomes the low order accuracy in traditional finite element method caused by the difference discretization in time. This method has high order accuracy in space and time directions, good dissipation on unstructured mesh, and unconditional stability. Thus, it becomes an efficiency method for the problems dependent on time. It has been successfully used in the fields of fluid mechanics, heat conduction, elastic dynamics, and structural mechanics.Prior to the work of [2], Lesaint and Raviart [3] and Jamet [4] used such time discontinuous Galerkin finite element method as mentioned above to study the partial differential equations, respectively. After that, many works were analyzed about both the theoretical results and use of the time discontinuous Galerkin finite element method, such as the works of Eriksson, Johnson, Thomée, and Babuska (cf. [5–9]).Sobolev equations are one of the important partial differential equations in practical use. They arise in the percolation theory of fluid through the crack of rock [10], the problem of moisture migration in soil [11], the problem of heat conduction between different media [12, 13], and so on. The existence and uniqueness of the solution of the Sobolev equation were discussed in [14, 15]. Numerical treatments for Sobolev equations can be found in many papers. The finite element method was discussed for the semilinear Sobolev equations in [16] and the nonlinear Sobolev equations in [17–20]. The differences-streamline diffusion method and DG finite element method were also discussed for Sobolev equations in [21–23], respectively.In this paper, we will establish a time discontinuous Galerkin finite element scheme for the quasi-linear Sobolev equations. That is to say, the approximate solution will be sought as a piecewise polynomial in time variable of degree at mostq
-
1 with coefficients in finite element space, which is not necessarily continuous at the nodes of the partition for the time interval. More attentions will be paid to treating a damping term u
x
x
t, which is a distinct character of Sobolev equations different from parabolic equation. To our knowledge, this paper appears to be the first trial to approximate quasi-linear Sobolev equations by using the time discontinuous Galerkin finite element method.The rest of this paper is organized as follows. In Section2, we briefly introduce the quasi-linear Sobolev equations and some assumptions. In Section 3, we present the time discontinuous Galerkin finite element scheme. In Section 4, we prove the existence and uniqueness of the approximate solution by use of Brouwer’s fixed point theorem. In Section 5, an optimal L
∞
(
0
,
T
;
H
1
(
Ω
)
)-norm error estimate is derived. Finally, we describe conclusions and perspectives in Section 6.


## 2. Mathematical Model

We consider the following quasi-linear Sobolev equations: to findu
=
u
(
x
,
t
) such that(1)∂u∂t-∑i,j=1d∂∂xiaijx,t,u∂u∂xj+bijx∂ut∂xj=fx,t,u,x,t∈Ω×0,T,u=0,x,t∈∂Ω×0,T,ux,0=u0x,x∈Ω,t=0,where Ω is a convex domain in R
d (d
≤
3) with a smooth boundary ∂
Ω.We consider the coefficients in (1) with the following conditions (I):(i)
f
∈
L
2
(
0
,
T
;
L
2
(
Ω
)
), u
0
∈
H
0
1
(
Ω
).
(ii)
Assumea
i
j
(
x
,
t
,
u
)
=
a
j
i
(
x
,
t
,
u
), b
i
j
(
x
)
=
b
j
i
(
x
), ∀
i
,
j
∈
1
,
…
,
d, x
∈
Ω, t
∈
(
0
,
T
]. There exist positive constants a
0, a
1, b
0, and b
1 satisfying(2)a0∑i=1dξi2≤∑i,j=1daijx,t,uξiξj≤a1∑i=1dξi2,∀ξ∈Rd,x∈Ω,t∈0,T,b0∑i=1dξi2≤∑i,j=1dbijxξiξj≤b1∑i=1dξi2,∀ξ∈Rd,x∈Ω,t∈0,T.
(iii)
Assume thata
i
j
(
x
,
t
,
z
) and f
(
x
,
t
,
z
) satisfy the uniform Lipschitz condition with respect to the variable z. Let a
i
j
(
x
,
t
,
z
) have uniform bounded derivatives of order up to 1 and 2 with respect to the variable t and variable z, respectively.For the purpose of theoretical analyses, we need the following regularity assumptions on the solution of (1):(II)(3)u∈L∞0,T;H01Ω∩L20,T;Hr+1Ω,r≥1,ut∈L∞0,T;Hr+1Ω∩L20,T;Hr+1Ω,Dtqu∈L∞0,T;H1Ω∩L20,T;H1Ω,where u
t
=
∂
u
/
∂
t and D
t
q denotes the qth-order derivative with respect to t, q
≥
1.


## 3. Time Discontinuous Galerkin Finite Element Scheme

In this section, we establish the approximate scheme based on the time discontinuous Galerkin method for (1).First, we introduce two bilinear forms(4)Au;w,v=∑i,j=1daijx,t,u∂w∂xi,∂v∂xj,∀u,w,v∈H01Ω,Bw,v=∑i,j=1dbijx∂w∂xi,∂v∂xj,∀u,w,v∈H01Ω.With the above notations, the variational form of (1) can be rewritten as follows: to find u
(
t
)
∈
H
0
1
(
Ω
) such that(5)∫0Tut,v+Au;u,v+But,vdt=∫0Tfu,vdt,∀vt∈H01Ω,ux,0=u0x,∀x∈Ω.Let0
=
t
0
<
t
1
<
⋯
<
t
N
=
T be an unnecessarily uniform subdivision of [
0
,
T
], and (6)In=tn,tn+1,tn+1-tn=τn,τ=maxnτn,∀0≤n≤N-1.LetT
h
=
{
K
} be a partition of Ω, let h
K be the diameter of element K, and h
=
m
a
x
K
∈
T
h
h
K. Define S
h
,
0
⊂
H
0
1
(
Ω
) as a finite element space with index r (cf. [24]). We want to find an approximate solution of (1) belonging to the space(7)Vh,τ=vh:vhIn=∑j=0q-1ωh,jtj,ωh,j∈Sh,0.In other words, a function in V
h
,
τ reduces to a polynomial of degree at most q
-
1 in t on each interval I
n with coefficients in S
h
,
0. Note that these functions in V
h
,
τ are allowed to be discontinuous at the nodes t
n, n
=
0
,
…
,
N
-
1, and are taken to be continuous to the left side of the nodes. For U
∈
V
h
,
τ, we denote U
n and U
+
n as the value of U and its limit from above t
n, respectively. We write V
h
,
τ
n for the restrictions to I
n of the function in V
h
,
τ.Our time discontinuous Galerkin finite element scheme is to findU
∈
V
h
,
τ such that(8)∑n=0N-1Un+1,vn+1-Un,v+n-∫InU,vtdt+∫InAU;U,vdt+BUn+1,vn+1-BUn,v+n-∫InBU,vtdt=∑n=0N-1∫InfU,vdt,∀v∈Vh,τ,with an initial value(9)U0=u0,h.Here, u
0
,
h is a certain approximation to u
0 in S
h
,
0.Noticing that there exist(10)-∑n=0N-1∫InU,vtdt=∑n=0N-1∫nUt,vdt-Un+1,vn+1+U+n,v+n,-∑n=0N-1∫InBU,vtdt=∑n=0N-1∫InBUt,vdt-BUn+1,vn+1+BU+n,v+n,we can rewrite (8) as(11)∑n=0N-1∫InUt,v+AU;U,v+BUt,vdt+U+n-Un,v+n+BU+n-Un,v+n=∑n=0N-1∫InfU,vdt,∀v∈Vh,τn,n=0,1,…,N-1.Hence, on each time intervalI
n, we have a local scheme: to find U
∈
V
h
,
τ
n, n
=
0,1
,
…
,
N
-
1, such that(12)∫InUt,v+AU;U,v+BUt,vdt+U+n,v+n+BU+n,v+n=Un,v+n+BUn,v+n+∫InfU,vdt,∀v∈Vh,τn.Remark 1.
From scheme (12), we can notice that the solution on I
n can be determined once U
n and U
+
n are given. In fact, the solution is defined on I
n by q elements in S
h
,
0, or q
N
h scalars if the dimension of S
h
,
0 is N
h, and the number of equations, which is equal to the dimension of  V
h
,
τ
n, is also q
N
h. Therefore, the main advantage of our scheme is that the approximate solution can be solved in each time interval gradually.


## 4. Existence and Uniqueness of the Solution

In this section, we prove the existence and uniqueness of solution (12). We can obtain the following theorem.Theorem 2.
Whenτ
n is small enough, the solution of (12) exists uniquely.Proof.
Let us introduce a projectionT
:
V
h
,
τ
n
↦
V
h
,
τ
n by(13)∫InYt,v+AX;Y,v+BYt,vdt+Y+n-Un,v+n+BY+n-Un,v+n=∫InfX,vdt,∀v∈Vh,τn,where Y
=
T
X, X
∈
V
h
,
τ
n.
Obviously, for anyX
∈
V
h
,
τ
n, the above equation has a unique solution so that the projection T is uniquely identified in V
h
,
τ
n. If T has a fixed point, X
∗
=
T
X
∗, then U
=
X
∗ is the solution of (12). If T has a unique fixed point, the solution of (12) is unique.
Now, we begin to prove thatT has a unique fixed point by three steps. For a real number R
>
0, we define a set(14)DR=X∈Vh,τn,∫InX-Un2+b0∇X-Un2dt≤R.It is easy to see that D
R is a closed set in V
h
,
τ
n.
Step 1 (we prove that when
τ
n
is small enough,
T
maps
D
R
onto itself). (
1
) Taking v
=
Y
-
U
n in (13), we can obtain(15)12∫InddtY-Un2dt+∫InAX;Y-Un,Y-Undt+∫InAX;Un,Y-Undt+12∫Inddt∇Y-Unb2dt+Y+n-Un2+∇Y+n-Unb2=∫InfX-fUn,Y-Undt+∫InfUn,Y-Undt,where the norm v
b
2
≔
(
b
(
x
)
v
,
v
), ∀
v
∈
V
h
,
τ
n.
Applying theɛ-inequality, we find(16)∫InAX;Un,Y-Undt≤12a0∫Ina12∇Un2dt+a02∫In∇Y-Un2dt≤τna122a0∇Un2+a02∫In∇Y-Un2dt,∫InfX-fUn,Y-Undt+∫InfUn,Y-Undt≤2ɛ∫InY-Un2dt+14ɛ∫InM2X-Un2+fUn2dt≤2ɛ∫InY-Un2dt+M24ɛ∫InX-Un2dt+τn4ɛfUn2,where M is the Lipschitz constant of f
(
x
,
t
,
z
) with respect to z.
Combining (15) with (16) shows(17)12Yn+1-Un2+Y+n-Un2+a02∫In∇Y-Un2dt+12∇Yn+1-Unb2+∇Y+n-Unb2≤τna122a0∇Un2+M24ɛ∫InX-Un2dt+τn4ɛfUn2+2ɛ∫InY-Un2dt.
(2) Takingv
=
(
t
-
t
n
)
Y
t in (13), since v
+
n
=
0, we have(18)∫Int-tnYt2+a02ddt∇Y2+∇Ytb2dt≤∫InfX,t-tnYtdt.
Notice that there exists(19)∫Int-tnddt∇Y2dt=τn∇Yn+12-∫In∇Y2dt.Hence, inserting (19) into (18) yields(20)∫Int-tnYt2+∇Ytb2dt+a0τn2∇Yn+12≤a02∫In∇Y2dt+∫InfX-fUn,t-tnYtdt+∫InfUn,t-tnYtdt.
Using theɛ-inequality to (20), we obtain(21)∫Int-tnYt2+b0∇Yt2dt+a0τn2∇Yn+12≤a02∫In∇Y2dt+ɛ∫Int-tnYt2dt+M24ɛ∫Int-tnX-Un2dt+ɛ∫Int-tnYt2dt+τn24ɛfUn2.
Therefore, by takingɛ
=
1
/
4 in (21), the following estimate holds:(22)12∫Int-tnYt2+b0∇Yt2dt+a0τn2∇Yn+12≤a02∫In∇Y2dt+M2τn∫InX-Un2dt+τn2fUn2.
(3) It is easy to see that there exists(23)∫InY-Un2dt≤2τnYn+1-Un2+τn∫InYt2dt,∫In∇Y-Un2dt≤2τn∇Yn+1-Un2+τn∫In∇Yt2dt.
AsY
t
2 and ∇
Y
t
2 are both polynomials in time of degree 2
(
q
-
2
) in I
n, and the norms are equivalent on finite dimensional space, we know that there exists a constant C
0
>
0, such that(24)τn∫InYt2dt≤C0∫Int-tnYt2dt,τn∫In∇Yt2dt≤C0∫Int-tn∇Yt2dt.
Then, it follows from (23) and (24) that(25)∫InY-Un2+b0∇Y-Un2dt≤2τnYn+1-Un2+τn∫InYt2dt+b0∇Yn+1-Un2+τnb0∫In∇Yt2dt≤2τnYn+1-Un2+b0∇Yn+1-Un2+C0∫Int-tnYt2+b0∇Yt2dt.Furthermore, (25) can be amplified as(26)∫InY-Un2+b0∇Y-Un2dt≤2τnYn+1-Un2+b0∇Yn+1-Un2+a0∫In∇Y-Un2dt+C0∫Int-tnYt2+b0∇Yt2dt+a0C0τn∇Yn+12.
Putting (17) and (22) into the right-hand side of (26) and using(27)∫In∇Y2dt≤2∫In∇Y-Un2dt+2τn∇Un2,we have(28)∫InY-Un2+b0∇Y-Un2dt≤2τn4ɛ∫InY-Un2dt+2C0a0∫In∇Y-Un2dt+2τna12a0+2C0a0τn∇Un2+M22ɛ+2C0M2τn∫InX-Un2dt+τn2ɛ+2C0τn2fUn2.
Ifτ
n is taken to be small enough to satisfy (i) m
a
x
{
8
τ
n
ɛ
,
4
τ
n
(
C
0
a
0
/
b
0
)
}
<
1
/
2 and (ii) the last term on the right-hand side of (28) <
R
/
2, we derive(29)∫InY-Un2+b0∇Y-Un2dt≤R;that is, T
(
D
R
)
⊂
D
R.
Step 2 (we prove that when
τ
n
is small enough,
T
is continuous on
D
R
). For any point X
0
∈
D
R, X
∈
D
R, denote Y
0
=
T
X
0, Y
=
T
X. Then, from (13) we have(30)∫InY-Y0t,v+AX;Y-Y0,v+BY-Y0t,vdt+Y-Y0+n,v+n+BY-Y0+n,v+n=∫InfX-fX0,v-AX;Y0,v+AX0;Y0,vdt.
Takingv
=
Y
-
Y
0 in (30) and with similar analyses in Step 1, we can find that when τ
n is small enough, there exists(31)limX→X0∫InY-Y02+b0∇Y-Y02dt=0.This means T is continuous on D
R.
Step 3. The above proofs show that T is a contraction mapping on D
R. Using Brouwer’s fixed point theorem, we know that T has a unique fixed point in D
R; that is, X
∗
=
T
X
∗. Therefore, the existence and uniqueness of the solution U
=
X
∗ of (12) have been proved.
Thus, the proof of Theorem2 is completed.


## 5.L
∞
(
0
,
T
;
H
1
(
Ω
)
)-Norm Error Estimate

In order to derive an optimalL
∞
(
0
,
T
;
H
1
(
Ω
)
)-norm error estimate, it is necessary to refer to a nonstandard elliptic projection P
1
:
u
∈
[
0
,
T
]
→
P
1
u
∈
S
h
,
0 from [17] such that(32)Au;P1u-u,v+BP1u-ut,v+λP1u-ut,v=0,∀v∈Sh,0,where constant λ
>
0. Also, we have the following lemmas about projection P
1.Lemma 3 (see [17]).
There exists a unique solutionP
1
u
∈
S
h
,
0 for any λ
>
0.Lemma 4 (see [17]).
Assume thatu
0
,
h
-
u
0
+
h
u
0
,
h
-
u
0
1
≤
C
h
r
+
1
u
0
r
+
1 and u, u
t
∈
H
r
+
1
(
Ω
), u, ∇
u, u
t, and ∇
u
t are bounded. Then, there exists a positive constant C
=
C
(
u
) such that(33)P1u-u+h∇P1u-u≤Chr+1uHr+1,P1u-ut+h∇P1u-ut≤Chr+1utHr+1.Lemma 5 (see [17]).
Assume thatu, u
t
∈
H
r
+
1
(
Ω
) and u, ∇
u, u
t, and ∇
u
t are bounded. There exists a positive constant C
=
C
(
u
) such that(34)P1uL∞+∇P1uL∞+P1utL∞+∇P1utL∞≤Cu.Lemma 6 (see [17]).
Letπ
=
P
1
u
-
u. There exists a positive constant C such that(35)πL20,T;L2Ω≤Chr+1uL20,T;Hr+1Ω,πtL20,T;L2Ω≤Chr+1utL20,T;Hr+1Ω.
LetU
~
∈
V
h
,
r
n be the interpolation of P
1
u with respect to t satisfying(36)U~tn+1-jq-1τn=P1utn+1-jq-1τn,0≤j≤q-1.By theLagrange interpolation theory ([24]), the following lemma holds.Lemma 7.
Letρ
=
U
~
-
P
1
u, where U
~ satisfies (36). Then, there exist(37)supInDtiρHs≤Cτnj-isupInDtjuHs,∫InDtiρHs2dt≤Cτn2j-i∫InDtjuHs2dt,where 0
≤
i
≤
q
-
1, i
+
1
≤
j
≤
q, and -
1
≤
s
≤
1; D
t
i denotes the ith-order derivative with respect to t; C is a positive constant depending only on q.We begin to analyze the error estimate aboutu
-
U. From (5) and (12), we can obtain the error equation(38)∫InUt-ut,v+AU;U,v-Au;u,v+BUt-ut,vdt+U+n,v+n+BU+n,v+n=Un,v+n+BUn,v+n+∫InfU-fu,vdt,∀v∈Vh,τn.Definingθ
=
U
-
U
~, π
=
P
1
u
-
u, then we have U
-
u
=
θ
+
ρ
+
π. Taking them into (38), we derive(39)∫Inθt+ρt+πt,v+AU;θ+ρ+π,v+AU;u,v-Au;u,v+Bθt+ρt+πt,vdt+θ+n-θn,v+n+Bθ+n-θn,v+n=∫InfU-fu,vdt,∀v∈Vh,τn.Applying elliptic projection (32), we find(40)AU;π,v+AU;u,v-Au;u,v+Bπt,v=AU;P1u,v-Au;u,v+Bπt,v=AU;P1u,v-Au;P1u,v+Au;P1u-u,v+Bπt,v=AU;P1u,v-Au;P1u,v-λπt,v.Hence, (39) can be rewritten to(41)∫Inθt,v+AU;θ,v+Bθt,vdt+θ+n-θn,v+n+Bθ+n-θn,v+n=∫Inρ,vt+λ-1πt,v-AU;ρ,v+Bρ,vtdt+∫InAu;P1u,v-AU;P1u,vdt+∫InfU-fu,vdt,∀v∈Vh,τn.From the two inequalities(42)θt≤θn+1+∫Inθtdt,∇θt≤∇θn+1+∫In∇θtdt,we know that(43)θt2≤2θn+12+τn∫Inθt2dt,∇θt2≤2∇θn+12+τn∫In∇θt2dt.So, it remains to estimateθ
n
+
1
1 and ∫
I
n
∇
θ
t
d
t. From now on, we use notations ·
=
·
L
2
(
Ω
) and ·
1
=
·
H
1
(
Ω
).Lemma 8.
Letξ
∈
V
h
,
τ
n, ζ, ζ
t
∈
L
2
(
I
n
,
H
1
(
Ω
)
) satisfying ζ
n
=
ζ
n
+
1
=
0, η, η
t
∈
L
2
(
I
n
,
L
2
(
Ω
)
). Define ξ
+
ζ
+
η
=
U
-
u to satisfy(44)∫Inξt,v+AU;ξ,v+Bξt,vdt+ξ+n-ξn,v+n+Bξ+n-ξn,v+n=∫Inζ,vt+λ-1ηt,v-AU;ζ,v+Bζ,vtdt+∫InAu;P1u,v-AU;P1u,vdt+∫InfU-fu,vdt,∀v∈Vh,τn.Then, we can have the following estimates:(45)∫Inξt2+b0∇ξt2dt≤C∫Inζt12+ζ12+ηt2+ξ12+η2dt, (46)ξn+12+b0∇ξn+12+a0∫In∇ξ2dt≤ξn2+b1∇ξn2+Cɛ∫Inζt12+ζ12+η2+ηt2dt,(47)ξn+112≤ξ012+∫Inζt12+ζ12+η2+ηt2dt.Proof.
(1) Taking v
=
(
t
-
t
n
)
ξ
t in (44), since v
+
n
=
0, we can get(48)∫Int-tnξt,ξt+AU;ξ,ξt+Bξt,ξtdt=∫Int-tn-ζt,ξt+λ-1ηt,ξt-AU;ζ,ξt-Bζt,ξtdt+∫Int-tn·Au;P1u,ξt-AU;P1u,ξtdt+∫Int-tnfU-fu,ξtdt.Further, we have(49)∫Int-tnξt2+∇ξtb2dt≤12∫Int-tn·ξt2+∇ξtb2dt+C∫Int-tn·ζt12+∇ζ2+ηt2+ξ12+ζ2+η2dt.
Becauseξ
t
2 and ∇
ξ
t
2 are both polynomials in t of degree at most 2
(
q
-
2
), we have two inverse inequalities(50)τn∫Inξt2dt≤C∫Int-tnξt2dt,(51)τn∫In∇ξt2dt≤C∫Int-tn∇ξt2dt.Combining the above inequalities with (49), we obtain (52)∫Inξt2+b0∇ξt2dt≤C∫Inζt12+ζ12+ηt2+ξ12+η2dt.This is just estimate (45).
(2) Now, we go to prove estimate (46). Taking v
=
ξ into (44), we have (53)the  left-hand  side  of44≥∫Ind2dtξ2+a0∇ξ2+d2dt∇ξb2dt+12ξ+n2-ξn2+12∇ξ+nb2-∇ξnb2=12ξn+12-ξn2+a0∫In∇ξ2dt+12∇ξn+1b2-∇ξnb2,the  right-hand  side  of44≤ɛ∫Inξt2+∇ξ2dt+Cɛ∫Inζ12+η2+ηt2+ξ2dt.
From (53), we conclude that(54)12ξn+12-ξn2+a0∫In∇ξ2dt+12∇ξn+1b2-∇ξnb2≤ɛ∫Inξt2+∇ξ2dt+Cɛ∫Inζ12+η2+ηt2+ξ2dt.
Combining (54) with (45), taking ɛ to be small enough, and using the Gronwall lemma, we can derive estimate (46).
(3) Finally, we prove estimate (47). Define a bilinear form: ∀
U
,
w
,
v
∈
V
h
,
τ,(55)BnU;w,v=∑j=0n∫Ijwt,v+AU;w,v+Bwt,vdt+w+j-wj,v+j+b∇w+j-wj,∇v+j+w0,v+0+b∇w0,∇v+0.
From (44), we know that ξ, ζ, and η defined in Lemma 8 satisfy ∀
v
∈
V
h
,
τ
(56)BnU;ξ,v=∑j=0n∫Ijξt,v+AU;ξ,v+Bξt,vdt+ξ+j-ξj,v+j+b∇ξ+j-ξj,∇v+j+ξ0,v+0+b∇ξ0,∇v+0=∑j=0n∫Ijζ,vt+λ-1ηt,v-AU;ζ,vdt+∫IjAu;P1u,v-AU;P1u,vdt+∫IjfU-fu,vdt+ξ0,v+0+b∇ξ0,∇v+0.
Letv
h
∈
S
h
,
0. We introduce an auxiliary function β
∈
V
h
,
τ, which satisfies β
0
=
v
h and(57)BnU;β,v=cvh,v+0+b∇vh,∇v+0,∀v∈Vh,τ.Using (55), we know (58)∫Ijβt,v+AU;β,v+Bβt,vdt+β+n-βn,v+n+b∇β+n-βn,∇v+n=0,where β satisfies the conditions in Lemma 8.
Hence, from (45) and (46), we can derive the estimate(59)βn+12+∑j=0n∫Ijβt2+a0∇β2+b0∇βt2dt+b0∇βn+12≤vh2+b1∇vh2.
Now, we consider an adjoint problem of (1): to find β
∗
∈
V
h
,
τ such that(60)Bnu;w,β∗=vh,wn+1+b∇vh,∇wn+1.In other words, that β
∗ is the finite element solution in V
h
,
τ of (61)-∂β∗∂t-∑i,j=1d∂∂xiaijx,t,u∂β∗∂xj-bijx∂βt∗∂xj=0,x∈Ω,0<t<tn+1,β∗tn+1=vh,β∗=0,x∈∂Ω.
Similar to the derivation of result (59), we can bound the estimate(62)β+∗02+∑j=0n∫Ijβt∗2+b0∇βt∗2+a0∇β∗2dt+b0∇β+∗02≤vh2+b1∇vh2.
Takingw
=
ξ in (60), we have(63)ξn+1,vh+b∇ξn+1,∇vh=Bnu;ξ,β∗=BnU;ξ,β∗+Bnu;ξ,β∗-BnU;ξ,β∗.
We analyze the terms on the right-hand side of (63) term by term. Using (46) and (62), we have(64)Bnu;ξ,β∗-BnU;ξ,β∗=∑j=0n∫IjAu;ξ,β∗-AU;ξ,β∗dt≤Ca1∫0tn+1∇ξ∇β∗dt≤ξ02+b1∇ξ02+Cɛ∑j=0n∫Ijζt2+ζ12+η2+ηt2dt1/2·vh2+b1∇vh21/2.Using (56) and (62), we have(65)BnU;ξ,β∗≤ξ02+b1∇ξ02+Cɛ∑j=0n∫Ijζt2+ζ12+η2+ηt2dt1/2·β+∗02+b0∇β+∗02+∑j=0n∫Ijβt∗2+b0∇βt∗2+a0∇β∗2dt1/2≤ξ02+b1∇ξ02+Cɛ∑j=0n∫Ijζt2+ζ12+η2+ηt2dt1/2·vh2+b1∇vh21/2.
Substituting (64) and (65) into (63) yields(66)ξn+1,vh+b∇ξn+1,∇vh=BnU;ξ,β∗≤Cξ012+∑j=0n∫Ijζt12+ζ12+η2+ηt2dt1/2·vh1.By the arbitrariness of v
h, taking v
h
=
ξ
n
+
1 into (66), we get estimate (47).In order to obtain the optimal order estimate with respect tot, we still need the following lemma (Lemma 9).Lemma 9.
Under the conditions in Lemma8, one has(67)ξn+12+b0∇ξn+12+a0∫In∇ξ2dt≤ξn2+b1∇ξn2+C∫Inτnζt12+∇ζ2+η2+ηt2+τn-1ζ12+η2+ηt2dt.(68)τn∫Inξt2+b0∇ξt2dt≤Cξn2+b1∇ξn2+∫Inτnζt12+∇ζ2+η2+ηt2+τn-1ζ12+η2+ηt2dt.Proof.
(
1
) Taking v
=
(
t
-
t
n
)
ξ
t in (44), since v
+
n
=
0, we can get(69)∫Int-tnξt,ξt+Bξt,ξtdt=-∫Int-tn·AU;ξ,ξtdt+∫Int-tn·-ζt,ξt+λ-1ηt,ξt-AU;ζ,ξtdt+∫Int-tnAu;P1u,ξt-AU;P1u,ξtdt+∫Int-tnfU-fu,ξtdt.Noticing that(70)∫Int-tnAU;ξ,ξtdt=12τnAUn+1;ξn+1,ξn+1-12∫InAU;ξ,ξdt-12∫Int-tnAUt;ξ,ξdt,similar to the proof in step (1) of Lemma 8, we can derive(71)∫Int-tnξt,ξt+Bξt,ξtdt+τnAUn+1;ξn+1,ξn+1≤∫InAU;ξ,ξdt+C∫Int-tn·ζt12+∇ζ2+η2+ηt2+∇ξ2dt.
Combining inequalities (50), (51), and (71) and using the Gronwall lemma, we obtain(72)τn∫Inξt2+b0∇ξt2dt+τna0∇ξn+12≤C∫In∇ξ2+τnζt12+∇ζ2+η2+ηt2dt.
(
2
) Taking v
=
ξ into (44), similar to the analysis of (53) and (54), we also have(73)12ξn+12-ξn2+a0∫In∇ξ2dt+12∇ξn+1b2-∇ξnb2≤ɛ∫Inτnξt2+∇ξ2dt+Cɛ∫Inτn-1ζ12+η2+ηt2dt.
Combining estimates (73) and (72) together, letting ɛ be small enough, we derive estimate (67). Then, we insert the estimate of ∫
I
n
∇
ξ
2
d
t of (67) into (72) to get (68).Letξ
=
θ, ζ
=
ρ, and η
=
π
t. Using Lemmas 8 and 9, we deduce the following lemma.Lemma 10.
Under the conditions in Lemmas8 and 9, there exist(74)θn+112≤Cθ012+τ2q∫0tn+1Dtqu12dt+h2r+1∫0tn+1DtuHr+12dt,(75)θL∞0,T;H1Ω2≤Cθ0H1Ω2+h2r+1DtuL20,T;Hr+1Ω2+DtuL∞0,T;Hr+1Ω2+τ2qDtquL20,T;H1Ω2+DtquL∞0,T;H1Ω2+DtuL20,T;Hr+1Ω2.Proof.
It is easy to see that (74) follows directly from (68), Lemmas 4 and 7.
Notice that,∀
t
∈
I
n, there exists(76)θt12≤2θn+112+τn∫Inθt12dt.So combining the above estimate with (68) yields(77)θt12≤Cθn+112+θn12+∫Inτn-1ρ12+∇ρ2+τn-1πt2+τnρt12+τn∇ρ2+τnπt2dt.Using Lemmas 4 and 7, we have the following estimates: (78)∫Inτn-1ρ12+τnρt12dt≤Cτn2qsupInDtqu12,∫In∇ρ2+τn∇ρ2dt≤Cτn2qDtquL2In;H1Ω2,∫Inτn-1πt2dt≤Ch2r+1supInutHr+1Ω2,∫Inτnπt2dt≤C∫Inτnh2r+1utHr+1Ω2dt≤Cτn2q+h2r+12q/2q-1utL2In;Hr+1Ω2.Inserting the above estimates into (77) and using (74) again, we derive (75).Finally, using the triangle inequalityu
-
U
1
≤
θ
1
+
ρ
1
+
π
1, Lemmas 4 and 5, we can establish the following theorem.Theorem 11.
Letu be the solution of (1) and let U be the solution of (12), respectively. Assume that u is smooth enough to satisfy the required regularities in the analyses. Then, the following error estimate holds:(79)u-UL∞0,T;H1Ω≤Cu0-u0,hH1Ω+hruL∞0,T;Hr+1Ω+hr+1DtuL20,T;Hr+1Ω+DtuL∞0,T;Hr+1Ω+τqDtquL20,T;H1Ω+DtquL∞0,T;H1Ω+DtuL20,T;Hr+1Ω.
Furthermore, if an initial assumptionu
0
-
u
0
,
h
H
1
(
Ω
)
≤
C
h
r
u
0
H
r
+
1
(
Ω
) exists, we obtain(80)u-UL∞0,T;H1Ω≤Chr+τq.In other words, we get higher accuracy for both the space variable and the time variable. The results are consistent with [25].


## 6. Conclusions

The quasi-linear Sobolev equations (1) are solved by the time discontinuous Galerkin finite element methods. We presented the approximate scheme and proved the existence and uniqueness of the approximate solution by use of Brouwer’s fixed point theorem. We made the error analysis and derived the optimal L
∞
(
0
,
T
;
H
1
(
Ω
)
)-norm error estimates. We got higher accuracy for both the space variable and the time variable in this paper. In the forthcoming work, some numerical experiments will be considered to verify our analysis for the schemes.



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*Source: 985214-2015-08-20.xml*