Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 895187, 17 pages
doi:10.1155/2010/895187
Research Article
L2 -Error Estimates of the Extrapolated
Crank-Nicolson Discontinuous Galerkin
Approximations for Nonlinear Sobolev Equations
Mi Ray Ohm,1 Hyun Young Lee,2 and Jun Yong Shin3
1
Division of Information Systems Engineering, Dongseo University, 617-716 Busan, South Korea
Department of Mathematics, Kyungsung University, 608-736 Busan, South Korea
3
Division of Mathematical Sciences, Pukyong National University, 608-737 Busan, South Korea
2
Correspondence should be addressed to Hyun Young Lee, hylee@ks.ac.kr
Received 9 November 2009; Accepted 23 January 2010
Academic Editor: Jong Kim
Copyright q 2010 Mi Ray Ohm et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We analyze discontinuous Galerkin methods with penalty terms, namely, symmetric interior
penalty Galerkin methods, to solve nonlinear Sobolev equations. We construct finite element
spaces on which we develop fully discrete approximations using extrapolated Crank-Nicolson
method. We adopt an appropriate elliptic-type projection, which leads to optimal ∞ L2 error
estimates of discontinuous Galerkin approximations in both spatial direction and temporal
direction.
1. Introduction
Let Ω be an open bounded domain in Rd , d ≥ 2 with smooth boundary ∂Ω, and let 0 < T <
∞ be given. In this paper, we consider the problem of approximating ux, t satisfying the
following nonlinear Sobolev equations:
ut − ∇ · {au∇u bu∇ut } fu in Ω × 0, T ,
au∇u bu∇ut · n 0 on ∂Ω × 0, T ,
ux, 0 u0 x
1.1
on Ω,
where n denotes the unit outward normal vector to ∂Ω and u0 x is a given function defined
on Ω. The initial data u0 x, f, a, and b are assumed to be such that 1.1 admits a solution
sufficiently smooth to guarantee the convergence results to be presented below. For details
2
Journal of Inequalities and Applications
about the physical significance and various properties of existence and uniqueness of the
Sobolev equations, see 1–6.
Early, in 7–9 the authors constructed the Galerkin approximations to the solution of
1.1 with periodic boundary conditions in one-dimensional space and obtained the optimal
convergence in L2 normed space and superconvergence results. Recently, Lin 10 constructed
the Galerkin approximation of 1.1 with d 2 using Crank-Nicolson method and proved
the optimal convergence of error in L2 normed space. In 11 the authors constructed the
semidiscrete finite element approximations of 1.1 with nonlinear boundary condition and
obtained the optimal L2 -error estimates.
In this work we will approximate the solution of 1.1 using a discontinuous symmetric Galerkin method with interior penalties for the spatial discretization and extrapolated
Crank-Nicolson method for the time stepping. By implementing the extrapolated technique,
we induce the linear systems which can be solved explicitly, and thus obviate the order
reduction phenomenon which occurs when the system involved is nonlinear.
Compared to the classical Galerkin method, the discontinuous Galerkin method is
very well suited for adaptive control of error and can deliver high orders of accuracy when the
exact solution is sufficiently smooth. In 12 Rivière and Wheeler formulated and analyzed a
family of discontinuous methods to approximate the solution of the transport problem with
nonlinear reaction. They construct semidiscrete approximations which converge optimally
in h and suboptimally in r for the energy norm and suboptimally for the L2 norm. They
also constructed fully discrete approximations and proved the optimal convergence in the
temporal direction. Furthermore to solve reactive transport problems Sun and Wheeler in
13 analyzed three discontinuous Galerkin methods, namely, symmetric interior penalty
Galerkin method, nonsymmetric interior penalty Galerkin method, and incomplete interior
penalty Galerkin method. They obtained error estimates in L2 H 1 which are optimal in h
and nearly optimal in p and they developed a parabolic lift technique for SIPG which leads
to h-optimal and nearly p-optimal error estimates in L2 L2 and negative norms. Recently
in 14, 15 Sun and Yang adapted discontinuous Galerkin methods to nonlinear Sobolev
equations and obtained the optimal H 1 error estimates. The main object of this paper is
to obtain the optimal ∞ L2 error estimates in both the spatial direction and the temporal
direction by adopting an appropriate elliptic-type projection.
This paper is organized as follows. In Section 2, we introduce some notations and
preliminaries. In Section 3, we construct appropriate finite element spaces and define an
auxiliary projection and prove its convergence. In Section 4, we construct the extrapolated
discontinuous Galerkin fully discrete method which yields the second-order convergence in
the temporal direction. The corresponding error estimates of the approximate solutions are
also discussed.
2. Notations and Preliminaries
Let Eh {E1 , E2 , . . . , ENh } be a regular quasi-uniform subdivision of Ω where Ei is a triangle
or a quadrilateral if d 2 and Ei is a 3-simplex or 3-rectangle if d 3. Let hj diamEj and h max1≤j≤Nh hj . Here, the regular requirement is that there exists a constant ρ > 0 such
that each Ej contains a ball of radius ρhj . The quasi-uniformity requirement is that there is a
constant γ > 0 such that
h
≤γ
hj
for j 1, 2, . . . , Nh .
2.1
Journal of Inequalities and Applications
3
We denote the edges resp., faces for d 3 of Eh by {e1 , e2 , . . . , ePh , ePh 1 , . . . , eNh } where
ek has positive d − 1 dimensional Lebesgue measure, ek ⊂ Ω, 1 ≤ k ≤ Ph , and ek ⊂ ∂Ω,
Ph 1 ≤ k ≤ Nh . With each edge or face ek , we associate a unit normal vector nk to Ei if
ek ∂Ei ∩ ∂Ej and i < j. For k ≥ Ph 1, nk is taken to be the unit outward vector normal to
∂Ω.
For an s ≥ 0 and a domain E ⊂ Rd , we denote by H s E the Sobolev space of order
s equipped with the usual Sobolev norm · s,E . We simply write · s instead of · s,Ω if
E Ω and · E instead of · s,E if s 0. And also the usual seminorm defined on H s E is
denoted by | · |s,E .
Now for an s ≥ 0 and a given subdivision Eh , we define the following space:
H s Eh v ∈ L2 Ω|v|Ei ∈ H s Ei , i 1, 2, . . . , Nh .
2.2
For φ ∈ H s Eh with s > 1/2, we define the average function {φ} and the jump function φ
such that
1
1
φ|Ej |ek , ∀x ∈ ek , 1 ≤ k ≤ Ph ,
φ|Ei |ek φ 2
2
φ φ|Ei |ek − φ|Ej |ek , ∀x ∈ ek , 1 ≤ k ≤ Ph ,
2.3
where ek ∂Ei ∩ ∂Ej with i < j.
We associated the following broken norms with the space H s Eh :
φ
2
Nh
i1
φ
2
1
Nh i1
φ
2
1,Ei
φ
2
,
Ei
h2i ∇2 φ
2
Ei
σ
2.4
Jβ φ, φ ,
where
Ph
σk
Jβσ φ, ψ β
k1 |ek |
φ ψ ds,
β>0
2.5
ek
is an interior penalty term and σ is a discrete positive function that takes the constant value
σk on the edge ek and is bounded below by σ0 > 0 and above by σ ∗ > 0.
3. Finite Element Spaces and Convergence of Auxiliary Projection
For a positive integer r, we construct the following finite element spaces:
Dr Eh v ∈ L2 Ω|v|Ei ∈ Pr Ei , i 1, 2, . . . , Nh ,
where Pr Ei denotes the set of polynomials of degree less than or equal to r on Ei .
3.1
4
Journal of Inequalities and Applications
Now we state the following hp-approximation properties and trace inequalities whose
proofs can be found in 16, 17.
Lemma 3.1. Let Ej ∈ Eh and φ ∈ H s Ej . Then there exist a positive constant C depending on s,
γ, and ρ but independent of φ, r, and h and a sequence zhr ∈ Pr Ej , r 1, 2, . . . such that, for any
0 ≤ q ≤ s,
μ−q
φ−
zhr
q,Ej
≤C
hj
r s−q
φ
s,Ej
s ≥ 0,
φ
s,Ej
s>
1
,
2
φ
s,Ej
s>
3
,
2
μ−1/2
φ−
zhr
0,ej
≤C
hj
r s−1/2
μ−3/2
φ − zhr
1,ej
≤C
hj
r s−3/2
3.2
where μ minr 1, s and ej is an edge or a face of Ej .
Lemma 3.2. For each Ej ∈ Eh , there exists a positive constant C depending only on γ and ρ such that
the following trace inequalities hold:
φ
∂φ
∂nj
2
0,ej
≤C
1
φ
hj
2
≤C
0,ej
2
0,Ej
1
φ
hj
hj φ
2
1,Ej
∀φ ∈ H 1 Ej ,
,
2
1,Ej
hj φ
2
2,Ej
,
∀φ ∈ H Ej ,
3.3
2
where ej is an edge or a face of Ej and nj is the unit outward normal vector to Ej .
Now we introduce the following bilinear mappings Aρ; ·, · and Bρ; ·, · defined on
H s Eh × H s Eh as
Ph
A ρ; φ, ψ a ρ ∇φ, ∇ψ −
k1
Ph
a ρ ∇φ · nk ψ −
ek
k1
ek
a ρ ∇ψ · nk φ Jβσ φ, ψ ,
Ph Ph B ρ; φ, ψ b ρ ∇φ, ∇ψ −
b ρ ∇φ · nk ψ −
b ρ ∇ψ · nk φ Jβσ φ, ψ .
k1
ek
k1
ek
3.4
Using the bilinear mappings A and B, we construct the weak formulation of problem 1.1 as
follows:
ut t, v Aut; ut, v But; ut t, v fut, v ,
∀v ∈ H s Eh .
3.5
Journal of Inequalities and Applications
5
Now for a λ > 0 we define the following bilinear forms Aλ ρ; ·, · and Bλ ρ; ·, · on H s Eh ×
H s Eh such that
Aλ ρ; φ, ψ A ρ; φ, ψ λ φ, ψ ,
Bλ ρ; φ, ψ B ρ; φ, ψ λ φ, ψ .
3.6
Aλ and Bλ satisfy the following boundedness and coercivity properties, respectively. The
proofs can be found in 18, 19.
Lemma 3.3. For a λ > 0, there exists a constant C > 0 satisfying
Aλ ρ; φ, ψ ≤ C φ
Bλ ρ; φ, ψ ≤ C φ
1
ψ
1
1
ψ
1
,
∀φ, ψ ∈ H s Eh .
3.7
Lemma 3.4. For a λ > 0, there exists a constant c > 0 satisfying
∼
Aλ ρ; φ, φ ≥ c φ
∼
Bλ ρ; φ, φ ≥ c φ
∼
2
1
2
,
1
∀φ ∈ H s Eh .
3.8
Wheeler 20 introduced an elliptic projection to prove the optimal L2 -error estimates
for Galerkin approximation to parabolic differential equations. Adopting this idea we
construct a projection u
t : 0, T → Dr Eh such that
Aλ u; u − u
t , v 0
, v Bλ u; ut − u
u0, v u0, v.
∀v ∈ Dr Eh ,
3.9
By Lemmas 3.3 and 3.4, u
t is well defined.
4. The Optimal ∞ L2 Error Estimates of
Fully Discrete Approximations
In this section we construct fully discrete discontinuous Galerkin approximations using
extrapolated Crank-Nicolson method and prove the optimal convergence in L2 normed
space.
For a positive integer N > 0 we let Δt T/N and for 0 ≤ j ≤ N and we define tj jΔt and gj gx, tj . For 0 ≤ j ≤ N−1, we define ∂t gj gj1 −gj /Δt, tj1/2 1/2tj tj1 ,
and gj1/2 1/2gtj gtj1 .
6
Journal of Inequalities and Applications
The extrapolated Crank-Nicolson discontinuous Galerkin approximation {Uj }N
j0 ⊂
Dr Eh is defined by
∂t Uj , v A EUj : Uj1/2 , v B EUj ; ∂t Uj , v f EUj , v ,
∀v ∈ Dr Eh ,
4.1
where EUj 3/2Uj − 1/2Uj−1 , Uj1/2 1/2Uj Uj1 .
To apply 4.1, we need two initial stages U0 and U1 to be defined in the following:
∂t U0 , v AU1/2 ; U1/2 , v BU1/2 ; ∂t U0 , v fU1/2 , v ,
0,
U0 u
4.2
where U1/2 1/2U0 U1 .
To prove the optimal convergence of utj − Uj in L2 normed space we denote ηx, t x, tj − Uj x, j 0, 1, . . ., N.
ux, t − u
x, t and ξx, tj u
Now we state the following approximations for η whose proofs can be found in 18,
19.
Theorem 4.1. If ut ∈ L2 H s and u0 ∈ H s , then there exists a constant C independent of h and Δt
satisfying
i |ηt | h|ηt |1 ≤ Chs ut H s u0 s ii |η| h|η|1 ≤ Chs ut L2 H s u0 s .
Theorem 4.2. If ut ∈ L2 H s , utt ∈ H s , uttt ∈ H s and u0 ∈ H s then there exists a constant C
independent of h and Δt satisfying
i |ηtt |1 ≤ Chs−1 {ut L2 H s utt s u0 s },
ii |ηttt |1 ≤ Chs−1 {ut L2 H s utt s uttt s u0 s }
provided that β ≥ 1/d − 1.
By simple computations and the applications of Theorem 4.2 we obtain the following
lemmas.
Lemma 4.3. If ρ satisfies
j − u
t tj1/2 Δtρj1/2 ,
∂t u
4.3
then there exists a constant C independent of h and Δt such that
ρj1/2
≤ CΔt u0 s ut L2 H s utt L∞ H s uttt L∞ H s ,
ρj1/2
≤ CΔt u0 s ut L2 H s utt L∞ H s uttt L∞ H s .
1
4.4
Journal of Inequalities and Applications
7
Consequently from Lemma 4.3 there exists a constant C independent of Δt and h such
that
ρj1/2
≤ CΔt,
ρj1/2
1
≤ CΔt
4.5
if u is sufficiently smooth.
Lemma 4.4. If rj1/2 u
tj1/2 − u
j1/2 , then there exists a constant C independent of h and Δt
such that
rj1/2
≤ CΔt2 u0 s ut L2 H s utt L∞ H s ,
rj1/2
≤ CΔt2 u0 s ut L2 H s utt L∞ H s .
1
4.6
Consequently from Lemma 4.4 there exists a constant C independent of Δt and h such
that
rj1/2
≤ CΔt2 ,
rj1/2
1
≤ CΔt2
4.7
if u is sufficiently smooth.
Lemma 4.5. If we let ϕj1/2 u
tj1/2 − 3/2
utj − 1/2
utj−1 , then there exists a constant
C independent of h and Δt such that
ϕj1/2
ϕj1/2
1
≤ CΔt2 u0 s ut L2 H s utt L∞ H s ,
≤ CΔt2 u0 s ut L2 H s utt L∞ H s utt L∞ H s .
4.8
Consequently from Lemma 4.5, we induce that there exists a constant C independent
of Δt and h such that
ϕj1/2
≤ CΔt2 ,
ϕj1/2
1
≤ CΔt2
4.9
if u is sufficiently smooth.
Theorem 4.6. For 0 < λ < 1 and δ > 0, if ut ∈ L∞ H s , utt ∈ L∞ H s , then there exists a constant
C > 0 independent of h and Δt such that for j 1, 2, . . ., N
u tj − Uj
≤ C hμ Δt2 u0 s ut L∞ H s ∇ut L∞ utt L∞ H s uttt L∞ H s 4.10
hold where s d/2 1 δ and μ minr 1, s.
8
Journal of Inequalities and Applications
Proof. From 4.1 and 1.1, we have
ut tj1/2 − ∂t Uj , v Aλ u tj1/2 ; u tj1/2 , v − Aλ EUj ; Uj1/2 , v
Bλ u tj1/2 ; ut tj1/2 , v − Bλ EUj ; ∂t Uj , v
f u tj1/2 − f EUj , v λ u tj1/2 − Uj1/2 , v λ ut tj1/2 − ∂t Uj , v .
4.11
By the notations of η and ξ, we get
ut tj1/2 − ∂t Uj ut tj1/2 − ∂t u
j ∂t u
j − ∂t Uj
ηt tj1/2 Δtρj1/2 ∂t ξj .
4.12
By the definition of η, we obtain
Aλ u tj1/2 ; u tj1/2 , v − Aλ EUj ; Uj1/2 , v
Aλ EUj ; ξj1/2 , v − Aλ EUj ; u
j1/2 , v Aλ u tj1/2 ; u tj1/2 , v
Aλ EUj ; ξj1/2 , v Aλ u tj1/2 ; η tj1/2 , v Aλ u tj1/2 ; u
tj1/2 − u
j1/2 , v
Aλ u tj1/2 ; u
j1/2 , v .
j1/2 , v − Aλ EUj ; u
4.13
From the definition of η, we have
Bλ u tj1/2 ; ut tj1/2 , v − Bλ EUj ; ∂t Uj , v
Bλ EUj ; ∂t ξj , v Bλ u tj1/2 ; ut tj1/2 , v − Bλ EUj ; ∂t u
j , v
Bλ EUj ; ∂t ξj , v Bλ u tj1/2 ; ut tj1/2 − ∂t u
j , v Bλ u tj1/2 ; ∂t u
j , v
− Bλ EUj ; ∂t u
j , v
Bλ EUj ; ∂t ξj , v Bλ u tj1/2 ; ηt tj1/2 − Δtρj1/2 , v Bλ u tj1/2 ; ∂t u
j , v
j , v .
− Bλ EUj ; ∂t u
4.14
Substituting 4.12–4.14 in 4.11 and choosing v ξj1/2 ∂t ξj imply that
∂t ξj , ξj1/2 ∂t ξj Aλ EUj ; ξj1/2 , ξj1/2 ∂t ξj Bλ EUj ; ∂t ξj , ξj1/2 ∂t ξj
− ηt tj1/2 Δtρj1/2 , ξj1/2 ∂t ξj − Aλ u tj1/2 ; η tj1/2 , ξj1/2 ∂t ξj
− Aλ u tj1/2 ; rj1/2 , ξj1/2 ∂t ξj − Aλ u tj1/2 ; u
j1/2 , ξj1/2 ∂t ξj
Aλ EUj ; u
j1/2 , ξj1/2 ∂t ξj − Bλ u tj1/2 ; ηt tj1/2 , ξj1/2 ∂t ξj
Journal of Inequalities and Applications
9
j , v
Bλ u tj1/2 ; Δtρj1/2 , ξj1/2 ∂t ξj − Bλ u tj1/2 ; ∂t u
Bλ EUj ; ∂t u
j , ξj1/2 ∂t ξj f u tj1/2 − f EUj , ξj1/2 ∂t ξj
λ u tj1/2 − Uj1/2 , ξj1/2 ∂t ξj λ ut tj1/2 − ∂t Uj , ξj1/2 ∂t ξj .
4.15
By Cauchy-Schwarz’s inequality clearly we have
1 ξj1
∂t ξj , ξj1/2 ≥
2Δt
2
−
2
ξj
4.16
.
By the definition of Aλ we have
Ph Aλ EUj ; ξj1/2 , ∂t ξj a EUj ∇ξj1/2 , ∇ ∂t ξj −
a EUj ∇ξj1/2 · nk ∂t ξj
k1
−
Ph a EUj ∇ ∂t ξj · nk ξj1/2 Jβσ ξj1/2 , ∂t ξj λ ξj1/2 , ∂t ξj
k1
≥ a0
ek
1 ∇ξj1
2Δt
λ
−
ek
1 ξj1
2Δt
2
2
−
∇ξj
−
ξj
2
1 σ
Jβ ξj1 , ξj1 − Jβσ ξj , ξj
2Δt
Ph −
a EUj ∇ξj1/2 · nk ∂t ξj
2
k1
ek
Ph a EUj ∇ ∂t ξj · nk ξj1/2 .
k1
ek
4.17
For the definition of Bλ we get
Bλ EUj ; ∂t ξj , ξj1/2
≥
1 2
2
2
a0 ∇ξj1 − ∇ξj
Jβσ ξj1 , ξj1 − Jβσ ξj , ξj λ ξj1 −
2Δt
Ph Ph −
b EUj ∇ξj1/2 · nk ∂t ξj −
b EUj ∇ ∂t ξj · nk ξj1/2 .
k1
ek
k1
ξj
2
ek
4.18
Applying 4.17 and 4.18 in 4.15 we conclude that
1 1 2λ ξj1
2Δt
∂t ξj
2
2
ξj
2
2a0
c ξj1/2
2
1
c ∂t ξj
∼
−
∼
∇ ξj1
2
1
2
−
∇ξj
2
2 J ξj1 , ξj1 − J ξj , ξj
10
Journal of Inequalities and Applications
≤ − ηt tj1/2 Δtρj1/2 , ξj1/2 ∂t ξj f u tj1/2 − f EUj , ξj1/2 ∂t ξj
λ u tj1/2 − Uj1/2 , ξj1/2 ∂t ξj λ ηt tj1/2 Δtρj1/2 , ξj1/2 ∂t ξj
− Aλ u tj1/2 ; rj1/2 , ξj1/2 ∂t ξj − Aλ u tj1/2 ; u
j1/2 , ξj1/2 ∂t ξj
Aλ EUj ; u
j1/2 , ξj1/2 ∂t ξj Bλ u tj1/2 ; Δtρj1/2 , ξj1/2 ∂t ξj
− Bλ u tj1/2 ; ∂t u
j , ξj1/2 ∂t ξj Bλ EUj ; ∂t u
j , ξj1/2 ∂t ξj
Ph Ph a EUj ∇ξj1/2 · nk ∂t ξj b EUj ∇ξj1/2 · nk ∂t ξj
ek
k1
Ph Ph a EUj ∇ ∂t ξj · nk ξj1/2 b EUj ∇ ∂t ξj · nk ξj1/2
ek
k1
ek
k1
ek
k1
10
Ii .
i1
4.19
For sufficiently small ε > 0 by applying Lemma 4.3 there exists a constant C > 0 such that
ηt tj1/2
ξj1/2 ∂t ξj
Δtρj1/2
2
2
2
Δt2 ρj1/2 ξj1 ξj
≤ C ηt tj1/2
|I1 | ≤
≤C
ηt tj1/2
2
Δt4 2
ξj1
2
ξj
2
ε ∂t ξj
2
ε ∂t ξj
2
4.20
.
Applying Lemmas 4.3 and 4.4, I2 can be estimated as follows:
ξj1/2 ∂t ξj
|I2 | ≤ C u tj1/2 − EUj
2
2
2
Δt4 ξj
ξj1 ≤ C η tj1/2
2
ξj−1
ε ∂t ξj
2
.
4.21
We obtain the following estimates of Ii for each 3 ≤ i ≤ 5 :
η tj1/2
η tj1/2
ηt tj1/2
|I3 | ≤ λ
≤C
|I4 | ≤ λ
≤C
ηt tj1/2
|I5 | ≤ C rj1/2
1
rj1/2
2
Δt4 ξj
Δt ρj1/2
2
Δt4 ξj1/2
1
ξj1/2
2
2
ξj1
∂t ξj
1
ξj1
ξj1/2
2
ξj1/2
2
ξj
ε ∂t ξj
∂t ξj
∂t ξj
2
.
ε ∂t ξj
≤ CΔt4 ε ξj1/2
4.22
2
.
2
1
ε ∂t ξj
2
.
1
Journal of Inequalities and Applications
11
From the definition of I6 , we can separate I6 as follows:
I6 a EUj − a u tj1/2
∇
uj1/2 , ∇ ξj1/2 ∂t ξj
−
Ph a EUj − a u tj1/2 ∇
uj1/2 · nk ξj1/2 ∂t ξj
ek
k1
Ph −
a EUj − a u tj1/2 ∇ ξj1/2 ∂t ξj · nk u
j1/2
ek
k1
3
4.23
I6i .
i1
By applying Lemma 4.5, I61 can be estimated in the following way:
uj1/2
I61 ≤ C ∇
≤C
∞
η tj1/2
η tj1/2
Δt2 Δt4 ξj−1
2
2
ξj−1
ξj
2
·
ξj
ε ξj1/2
∇ξj1/2
2
1
ε ∂t ξj
∇∂t ξj
2
.
1
4.24
Similarly there exists a constant C > 0 such that
I62 ≤ C
Ph
∇
uj1/2
k1
×
≤C
ξj1/2
Nh
0,ek
∇
uj1/2
× h−1/2 η tj1/2
h−1/2 ξj−1
η tj1/2
ε ξj1/2
2
1
∂t ξj
0,ek
ϕj1/2
0,ek
ξj
0,ek
ξj−1
0,ek
0,ek
∞,Ei
i1
≤C
η tj1/2
∞,ek
0,Ei
0,Ei
2
·
h1/2 ∇η tj1/2
ξj1/2
1
h2 ∇η tj1/2
ε ∂t ξj
2
∂t ξj
0,Ei
h−1/2 ϕj1/2
1
0,Ei
h−1/2 ξj
· hβd−1/2
Δt4 ξj
2
ξj−1
2
2
.
1
4.25
By applying the trace inequality we have
I63 ≤ C
Ph ∇ ξj1/2
k1
×
η tj1/2
0,ek
∞,ek
∇ ∂t ξj
ϕj1/2
0,Ei
0,ek
∞,ek
ξj
0,ek
ξj−1
0,ek
·
ηj1/2
0,ek
12
Journal of Inequalities and Applications
≤
Nh ∇ ξj1/2
∞,Ei
i1
×
η tj1/2
· h−1/2
≤C
Nh 0,Ei
ηj1/2
×
≤C
η tj1/2
0,Ei
η tj1/2
2
1
ε ξj1/2
2
h ∇ηj1/2
0,Ei
i1
∇∂t ξj
h−1/2
∞,Ei
h ∇η tj1/2
0,Ei
∇ ξj1/2
∇∂t ξj
0,Ei
ϕj1/2
0,Ei
ξj
0,Ei
ξj
0,Ei
ξj−1
0,Ei
0,Ei
ξj−1
0,Ei
0,Ei
0,Ei
h ∇η tj1/2
0,Ei
h2 ∇η tj1/2
ε ∂t ξj
ϕj1/2
2
Δt4 2
ξj
h−1−d/2s
2
ξj−1
2
.
1
4.26
From the estimation of I6i , 1 ≤ i ≤ 3, we have
|I6 | ≤ C
η tj1/2
3ε ξj1/2
2
1
2
h2 ∇η tj1/2
3ε ∂t ξj
2
ξj
2
ξj−1
2
Δt4
4.27
2
.
1
By applying Lemma 4.3 we obviously obtain
|I7 | ≤ CΔt ρj1/2
1
ξj1/2
1
∂t ξj
1
≤ CΔt4 ε ξj1/2
2
1
ε ∂t ξj
2
.
1
4.28
Now we can separate I8 as follows:
I8 ≤
b EUj − b u tj1/2
j , ∇ ξj1/2 ∂t ξj
∇ ∂t u
−
Ph j · nk ξj1/2 ∂t ξj
b EUj − b u tj1/2 ∇ ∂t u
k1
−
Ph j
b EUj − b u tj1/2 ∇ ξj1/2 ∂t ξj · nk ∂t u
k1
ek
3
j1
I8j .
ek
4.29
Journal of Inequalities and Applications
13
Since
∇∂t u
j ∞
1
Δt
tj1
∇
ut τdτ
≤ ∇
ut L∞ L∞ tj
4.30
∞
≤ C
ut L∞ H s ≤ C ut − u
t L∞ H s ut L∞ H s ,
I81 can be estimated as follows
∇ξj1/2 ϕj1/2 ξj ξj−1
j ∞ η tj1/2
I81 ≤ C ∇ ∂t u
2
2
2
2
2
≤ C η tj1/2
ξj
ξj−1 Δt4 ε ξj1/2 1 ε ∂t ξj 1 .
∇∂t ξj
4.31
We apply Lemma 3.2 to estimate I82 as follows:
I82 ≤ C
Ph
∇ ∂t u
j
k1
×
≤C
ξj1/2
Nh
0,ek
∇∂t u
j ∞,Ei
i1
η tj1/2
∞,ek
∂t ξj
· h−1/2
0,ek
≤C
η tj1/2
ε ξj1/2
2
1
2
η tj1/2
0,Ei
h2 ∇η tj1/2
ε ∂t ξj
0,ek
ξj
0,ek
ξj−1
0,ek
0,ek
ξj
ϕj1/2
2
0,Ei
h ∇η tj1/2
ξj−1
0,Ei
2
ξj
· h1/2
ξj−1
2
0,Ei
ϕj1/2
ξj1/2
1
0,ek
∂t ξj
1
Δt4
2
.
1
4.32
From the result of approximation of η of Theorem 4.1
I83 ≤ C
Ph ∇ξj1/2
k1
ηt tj1/2 Δtϕj1/2
·
≤C
∞,ek
∇∂t ξj
Nh
h−d/2
∇ξj1/2
0,Ei
i1
×
η tj1/2
× h−1/2 ·
0,Ei
∞,ek
0,ek
ϕj1/2
0,ek
ξj
0,ek
0,ek
∇∂t ξj
h ∇η tj1/2
ηt tj1/2
η tj1/2
0,Ei
0,Ei
0,Ei
h−1/2
ϕj1/2
h ∇ηt tj1/2
0,Ei
0,Ei
ξj
0,Ei
ξj−1
Δth ∇ϕj1/2
0,Ei
0,Ei
ξj−1
0,ek
14
Journal of Inequalities and Applications
≤C
≤C
ξj1/2
1
η tj1/2
2
1
ε ξj1/2
2
h ∇η tj1/2
η tj1/2
∂t ξj
1
h2 ∇η tj1/2
2
2
ξj
2
ξj−1
ϕj1/2
ξj
ξj−1
Δt4
2
.
1
ε ∂t ξj
4.33
Therefore we get
I8 ≤ C
η tj1/2
2
2
1
3ε ξj1/2
h2 ∇η tj1/2
3ε ∂t ξj
2
2
ξj
2
ξj−1
Δt4
4.34
2
.
1
Similarly, I9 and I10 are estimated as follows:
Ph I9 ≤
a EUj b EUj ∇ξj1/2 · nk ∂t ξj
ek
k1
≤C
Ph
∇ξj1/2
0,ek
∇ξj1/2
0,Ei
k1
≤C
Nh
i1
≤C
I10
2
∇ξj1
∂t ξj
0,ek
∂t ξj
1
2
∇ξj
ε ∂t ξj
4.35
2
.
1
Ph ≤
a EUj b EUj ∇ ∂t ξj · nk ξj1/2
ek
k1
≤C
Ph
∇∂t ξj k1
ξj1/2
0,ek
0,ek
≤ C Jβσ ξj1 , ξj1 Jβσ ξj , ξj ε ∂t ξj
2
.
1
Substituting the estimates of Ii , 1 ≤ i ≤ 10 into 4.19, we get
1 ξj1
2Δt
≤C
2
∂t ξj
−
ξj
2
ηt tj1/2
2
2
1
ξj1/2
2
∇ ξj1
2
2
1
∂t ξj
Δt4 h2 ∇η tj1/2
2
ξj1
∇ξj1
2
−
∇ξj
2
ξj
J ξj1 , ξj1 − J ξj , ξj
2
2
η tj1/2
2
ξj−1
2
|∇ξ|2 J ξj1 , ξj1 J ξj , ξj .
4.36
Journal of Inequalities and Applications
15
If we sum both sides of 4.36 from j 0 to N − 1, then we obtain
|ξN |2 − |ξ0 |2 |∇ξN |2 − |∇ξ0 |2 JξN , ξN Jξ0 , ξ0 N−1
2Δt
2
∂t ξj
2
1
ξj1/2
j0
⎧
⎨
N−1
η tj1/2
≤ C Δt
⎩
j0
Δt
N ξj
2
2
2
1
∂t ξj
ηt tj1/2
2
2
Δt4
4.37
⎫
⎬
2
∇ξj
J ξj , ξj h2 ∇η tj1/2
,
⎭
j0
which implies
|ξN |2 |∇ξN |2 JξN , ξN 2Δt
N−1
2
∂t ξj
2
1
ξj1/2
j0
∂t ξj
2
1
≤ |ξ0 |2 |∇ξ0 |2 Jξ0 , ξ0 CΔt
N−1
η tj1/2
2
ηt tj1/2
2
h2 ∇η tj1/2
2
Δt4
4.38
j0
CΔt
N ξj
2
∇ξj
2
J ξj , ξj ,
j0
where Δt is sufficiently small. By applying the discrete version of Gronwall’s inequality, we
have
|ξN |2 |∇ξN |2 JξN , ξN Δt
N−1
∂t ξj
2
j0
ξj1/2
2
1
2
1
∂t ξj
⎧
⎨
≤ C |ξ0 |2 |∇ξ0 |2 Jξ0 , ξ0 ⎩
Δt
N−1
j0
η tj1/2
2
4.39
ηt tj1/2
2
h2 ∇η tj1/2
2
Δt4
⎫
⎬
.
⎭
16
Journal of Inequalities and Applications
Therefore by applying the result of Lemma 4.7 we have
|ξ| ∞ L2 |∇ξ| ∞ L2 ≤ C hs Δt2 ,
|e| ∞ L2 ≤ C hs Δt2 ,
4.40
|e| ∞ H 1 ≤ C hs−1 Δt2 ,
which proves the optimal ∞ L2 error estimation of the fully discrete solutions.
Lemma 4.7 can be proved by the similar process of Theorem 4.6. as follows
Lemma 4.7. For 0 < λ < 1 and δ > 0, if ut ∈ L∞ H d/21δ , utt ∈ L∞ H d/21 , and h−d/2 Δt ≤ C0 ,
for some constant C0 then there exists a constant C > 0 independent of h and Δt
|ξ1 |L2 |∇ξ1 |L2 ≤ C hs Δt2 ,
|e1 |L2 ≤ C hs Δt2 .
4.41
Acknowledgment
This research was supported by Dongseo University Research Grants in 2009.
References
1 G. I. Barenblatt, I. P. Zheltov, and I. N. Kochina, “Basic concepts in the theory of seepage of
homogeneous liquids in fissured rocks,” Journal of Applied Mathematics and Mechanics, vol. 24, no.
5, pp. 1286–1303, 1960.
2 R. W. Carroll and R. E. Showalter, Singular and Degenerate Cauchy Problems, vol. 127 of Mathematics in
Science and Engineering, Academic Press, New York, NY, USA, 1976.
3 P. J. Chen and M. E. Gurtin, “On a theory of heat conduction involving two temperatures,” Zeitschrift
für Angewandte Mathematik und Physik, vol. 19, no. 4, pp. 614–627, 1968.
4 P. L. Davis, “A quasilinear parabolic and a related third order problem,” Journal of Mathematical
Analysis and Applications, vol. 40, no. 2, pp. 327–335, 1972.
5 R. E. Ewing, “Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations,”
SIAM Journal on Numerical Analysis, vol. 15, no. 6, pp. 1125–1150, 1978.
6 T. W. Ting, “A cooling process according to two-temperature theory of heat conduction,” Journal of
Mathematical Analysis and Applications, vol. 45, pp. 23–31, 1974.
7 D. N. Arnold, J. Douglas, Jr., and V. Thomée, “Superconvergence of a finite element approximation to
the solution of a Sobolev equation in a single space variable,” Mathematics of Computation, vol. 36, no.
153, pp. 53–63, 1981.
8 D. N. Arnold, “An interior penalty finite element method with discontinuous elements,” SIAM Journal
on Numerical Analysis, vol. 19, no. 4, pp. 742–760, 1982.
9 M. T. Nakao, “Error estimates of a Galerkin method for some nonlinear Sobolev equations in one
space dimension,” Numerische Mathematik, vol. 47, no. 1, pp. 139–157, 1985.
10 Y. Lin, “Galerkin methods for nonlinear Sobolev equations,” Aequationes Mathematicae, vol. 40, no. 1,
pp. 54–66, 1990.
11 Y. Lin and T. Zhang, “Finite element methods for nonlinear Sobolev equations with nonlinear
boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 165, no. 1, pp. 180–191,
1992.
Journal of Inequalities and Applications
17
12 B. Rivière and M. F. Wheeler, “Non conforming methods for transport with nonlinear reaction,” in
Fluid Flow and Transport in Porous Media: Mathematical and Numerical Treatment, vol. 295 of Contemporary
Mathematics, pp. 421–432, American Mathematical Society, Providence, RI, USA, 2002.
13 S. Sun and M. F. Wheeler, “Symmetric and nonsymmetric discontinuous Galerkin methods for
reactive transport in porous media,” SIAM Journal on Numerical Analysis, vol. 43, no. 1, pp. 195–219,
2005.
14 T. Sun and D. Yang, “A priori error estimates for interior penalty discontinuous Galerkin method
applied to nonlinear Sobolev equations,” Applied Mathematics and Computation, vol. 200, no. 1, pp.
147–159, 2008.
15 T. Sun and D. Yang, “Error estimates for a discontinuous Galerkin method with interior penalties
applied to nonlinear Sobolev equations,” Numerical Methods for Partial Differential Equations, vol. 24,
no. 3, pp. 879–896, 2008.
16 I. Babuška and M. Suri, “The h-p version of the finite element method with quasi-uniform meshes,”
RAIRO Modélisation Mathématique et Analyse Numérique, vol. 21, no. 2, pp. 199–238, 1987.
17 I. Babuška and M. Suri, “The optimal convergence rate of the p-version of the finite element method,”
SIAM Journal on Numerical Analysis, vol. 24, no. 4, pp. 750–776, 1987.
18 M. R. Ohm, H. Y. Lee, and J. Y. Shin, “Error estimates for discontinuous Galerkin method for nonlinear
parabolic equations,” Journal of Mathematical Analysis and Applications, vol. 315, no. 1, pp. 132–143,
2006.
19 M. R. Ohm, H. Y. Lee, and J. Y. Shin, “L2 -error analysis of discontinuous Galerkin approximations for
nonlinear Sobolev equations,” submitted.
20 M. F. Wheeler, “A priori L2 error estimates for Galerkin approximations to parabolic partial
differential equations,” SIAM Journal on Numerical Analysis, vol. 10, pp. 723–759, 1973.