Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 413718, 19 pages
doi:10.1155/2012/413718
Research Article
A Godunov-Mixed Finite Element
Method on Changing Meshes for the Nonlinear
Sobolev Equations
Tongjun Sun
School of Mathematics, Shandong University, Jinan 250100, China
Correspondence should be addressed to Tongjun Sun, tjsun@sdu.edu.cn
Received 1 September 2012; Accepted 14 November 2012
Academic Editor: Xinan Hao
Copyright q 2012 Tongjun Sun. 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.
A Godunov-mixed finite element method on changing meshes is presented to simulate the
nonlinear Sobolev equations. The convection term of the nonlinear Sobolev equations is
approximated by a Godunov-type procedure and the diffusion term by an expanded mixed
finite element method. The method can simultaneously approximate the scalar unknown and the
vector flux effectively, reducing the continuity of the finite element space. Almost optimal error
estimates in L2 -norm under very general changes in the mesh can be obtained. Finally, a numerical
experiment is given to illustrate the efficiency of the method.
1. Introduction
We consider the following nonlinear Sobolev equations:
ut − ∇ · ax, t, u∇ut bx, t, u∇u cx, t, u · ∇u fx, t, u,
ux, t 0,
x ∈ Ω, t ∈ 0, T ,
x ∈ ∂Ω, t ∈ 0, T ,
ux, 0 u0 x,
1.1
x ∈ Ω,
where Ω is a bounded subset of Rn n ≤ 3 with smooth boundary ∂Ω, u0 x and fx, t, u
are known functions, and the coefficients ax, t, u, bx, t, u, cx, t, u c1 x, t, u, c2 x, t, uT
satisfy the following condition:
∂ax, t, u ≤ K1 ,
0 < a0 ≤ ax, t, u ≤ a1 ,
∂t
0 < b0 ≤ bx, t, u ≤ b1 ,
2
|cx, t, u| c12 x, t, u c22 x, t, u ≤ c0 ,
ax, t, u, bx, t, u, cx, t, u, fx, t, u,
Abstract and Applied Analysis
∂cx, t, u ≤ K2 ,
∂u
∂cx, t, u
are Lipschitz continuous about u,
∂u
1.2
where a0 , a1 , b0 , b1 , c0 , K1 , and K2 are positive constants. We assume that ux, t satisfy the
smooth condition in the following analysis.
For time-changing localized phenomena, such as sharp fronts and layers, the finite
element method on changing meshes 1–3 is advantageous over fixed finite element method.
The reason is that the former treats the problem with the finite element method on space
domain by using different meshes and different basic functions at different time levels so that
it has the capability of self-adaptive local grid modification refinement or unrefinement
to efficiently capture propagating fronts or moving layers. The work 4 had combined this
method with mixed finite element method to study parabolic problems. In 5, an upwindmixed method on changing meshes was considered for two-phase miscible flow in porous
media.
Sobolev equations have important applications in many mathematical and physical
problems, such as the percolation theory when the fluid flows through the cracks 6, the
transfer problem of the moisture in the soil 7, and the heat conduction problem in different
materials 8. So there exists great and actual significance to research Sobolev equations.
Many works had researched on numerical treatments for Sobolev equations. More attentions
were paid for treating a damping term ∇ · a∇ut , which is a distinct character of Sobolev
equations different from parabolic equation. For example, time stepping Galerkin method
was presented for nonlinear Sobolev equations in 9, 10. In 11, 12, nonlinear Sobolev
equations with convection term were researched by using finite difference streamlinediffusion method and discontinuous Galerkin method, respectively. Two new least-squares
mixed finite element procedures were formulated for solving convection-dominated Sobolev
equations in 13.
Methods which combined Godunov-type schemes for advection with mixed finite
elements for diffusion were introduced in 14 and had been applied to flow problems in
reservoir engineering, contaminant transport, and computational fluid dynamics. Applications of these types of methods to single and two-phase flow in oil reservoirs were discussed
in 15, 16; application to the Navier-Stokes equations was given in 17. These methods
had proven useful for advective flow problems because they combined element-by-element
conservation of mass with numerical stability and minimal numerical diffusion. Dawson 18
researched advective flow problems in one space dimension by high-order Godunov-mixed
method. In 1993, he expanded this method to multidimensions 19 and presented three
variations. In these methods, advection was approximated by a Godunov-type procedure,
and diffusion was approximated by a low-order-mixed finite element method.
The object of this paper is to present a Godunov-mixed finite element method
on changing meshes for the nonlinear Sobolev equations. The convection term c · ∇u
of the nonlinear Sobolev equations is approximated by a Godunov-type procedure and
the diffusion term by an expanded mixed finite element method. This method can
simultaneously approximate the scalar unknown and the vector flux effectively, reducing
the continuity of the finite element space. We describe this method in Section 2. In
Section 3, we introduce three projections and a lemma. We derive almost optimal error
Abstract and Applied Analysis
3
estimates in L2 -norm under very general changes in the mesh in Section 4. In Section 5, we
present results of numerical experiment, which confirm our theoretical results.
Throughout the analysis, the symbol K will denote a generic constant, which is
independent of mesh parameters Δt and h and not necessarily the same at different
occurrences.
2. The Godunov-Mixed Method on Changing Meshes
At first we give some notation and basic assumptions. The usual Sobolev
spaces and norms
are adopted on Ω. The inner product on L2 Ω is denoted by f, g Ω fgdx. Define the
following spaces and norms:
H m div, Ω f fx , fy ; fx , fy , ∇ · f ∈ H m Ω ,
2 2
f2H m div fx m fy m ∇ · f2m ,
Hdiv, Ω H 0 div, Ω,
m ≥ 0,
L2 Ω
W
,
ϕ ≡ constant on Ω
2.1
V {v ∈ Hdiv; Ω | v · n 0 on ∂Ω, n is the unit outward norm to ∂Ω}.
∗
n
Let Δtn > 0 n 1, 2, . . . , N ∗ denote different time steps, tn nk1 Δtk , T N
n1 Δt ,
n
n
Δt maxn Δt . Assume that the time steps Δt do not change too rapidly; that is, we assume
there exist positive constants t∗ and t∗ which are independent of n and Δt such that
t∗ ≤
Δtn
≤ t∗ .
Δtn−1
2.2
For a given function gx, t, let g n gx, tn .
Assume Ω 0, 1 × 0, 1. At each time level tn , we construct a quasiuniform
rectangular partition Khn {ein } of Ω:
n
n
n
δxn : 0 x1/2
< x3/2
< · · · < xi1/2
< · · · < xInn 1/2 1,
n
n
n
< y3/2
< · · · < yj1/2
< · · · < yJnn 1/2 1.
δyn : 0 y1/2
2.3
n
n
n
n
Let hni,x xi1/2
− xi−1/2
, hni,y yj1/2
− yj−1/2
, hn maxi,j {hni,x , hni,y }, and h maxn hn . And
n
n
n
n
n
n
, hni1/2,x xi1
− xin , Δtn Ohn . Let
xi xi1/2 xi−1/2 /2 is the midpoint of xi−1/2 , xi1/2
n n
yj , hj1/2,y be defined analogously.
Let
k
k
n
n ∈ P
Bi,x
, i 1, . . . , I n ,
M−1
δxn f : f|Bi,x
k
M0k δxn M−1
δxn ∩ C0 0, 1,
2.4
n
n
n
n
xi−1/2
, xi1/2
and P k Bi,x
is the set of all polynomials of degree less than or
where Bi,x
0
n
δyn and M01 δyn .
equal to k defined on Bi,x . Similar definitions are given to M−1
4
Abstract and Applied Analysis
Then the lowest order Raviart-Thomas spaces Whn and Vhn are given by
0
0
δyn ,
Whn M−1
δxn ⊗ M−1
n
Vh,x
M01 δxn ⊗
0
M−1
δyn
,
n
Vh,y
n
n
Vhn Vh,x
× Vh,y
,
0
M−1
δxn ⊗
M01
δyn
2.5
.
That is, the space Whn is the space of functions which are constant on each element ein ∈ Khn ,
and Vhn is the space of vector valued functions whose components are continuous and linear
on each element ein ∈ Khn . The degrees of freedom of a function vn ∈ Vhn correspond to the
values of vn · γ at the midpoints of the sides of ein , where γ is the unit outward normal to ∂ein .
It is easy to see that Whn × Vhn ⊂ W × V and div Vhn Whn .
By introducing variables z −∇u, z bz azt , g cu c1 u, c2 uT g1 , g2 T , and
cx, t, u ∂c1 /∂u, ∂c2 /∂uT , we modify the first equation in 1.1 as
ut ∇ · z ∇ · g ucu · z fu.
2.6
Here we are using the so-called “expanded” mixed finite element method, proposed
by Arbogast et al. 20, which gives a gradient approximation z as well as an approximation
to the diffusion term z.
The weak form of 2.6 is
ut , w ∇ · z, w ∇ · g, w ucu · z, w fu, w ,
2.7
so that
ut , w ∇ · z, w ∇ · g, w ucu · z, w fu, w ,
z, v u, ∇ · v,
∀w ∈ W,
∀v ∈ V,
z, v buz auzt , v,
2.8
∀v ∈ V.
The Godunov-mixed method on changing meshes is presented as follows: at each time
level n, find Un ∈ Whn , Zn ∈ Vhn such that
Un − Rn Un−1 n
,w
Δtn
n
∇ · Zn , wn Un cUn · Z , wn
fUn , wn − ∇ · Gn−1 , wn ,
∀wn ∈ Whn ,
Abstract and Applied Analysis
Rn Un−1 − Un−1 , wn 0,
n
Z , vn Un , ∇ · vn ,
⎛
5
∀wn ∈ Whn ,
∀vn ∈ Vhn ,
n
n−1
Z − Rn Z
Z , v ⎝bU Z aU Δtn
n
n
n
n
n
n−1 , vn 0,
n−1 − Z
aUn Rn Z
⎞
n⎠
,v
,
∀vn ∈ Vhn ,
∀vn ∈ Vhn .
2.9
When different finite element spaces are used at time levels tn and tn−1 , the second and
n−1 } of the previous approximate
fifth equations of 2.9 give the L2 -projection {Rn Un−1 , Rn Z
n−1 n−1
solution {U , Z } into the current finite element space Whn × Vhn . This projection is used in
n }. If the finite element
the first and third equations of 2.9 as initial value to calculate {Un , Z
n
n−1
n n−1
n−1 .
n−1 Z
Un−1 , Rn Z
spaces are the same at time levels t and t , we know that R U
n
Let G in 2.9 be constructed by the Godunov method. We adopt the Godunov flux
19 Hw wL , wR for a given flux function ws:
L
Hw w , w
R
⎧
⎪
⎨ Lmin R ws,
if wL ≤ wR ,
⎪
⎩ Rmax L ws,
otherwise,
w ≤s≤w
w ≤s≤w
2.10
where wL , wR are the left and right states, respectively. It is easily seen that Hw is Lipschitz
in its arguments if w is Lipschitz in s and Hw is consistent, that is, Hw s, s ws.
n−1
Let G g 1 , g 2 , gi ci u, Gn−1
g n−1
1,i1/2,j , g 2,i,j1/2 . Now we give the
i1/2,j1/2
calculation of G by Hw by the following three steps.
First Step
We construct a piecewise linear function RUn−1 on each element ein−1 :
n−1
n−1
n−1
x − xi δx Ui,j
y − yj δy Ui,j
,
RUn−1 |ein−1 Ui,j
2.11
n−1
n−1
where δx Ui,j
and δy Ui,j
are x, y slopes. The x slope calculation can be performed as
n−1
δx Ui,j
⎧
⎨x Un−1 ,
i,j
⎩x− Un−1 ,
i,j
n−1 n−1 if x Ui,j
≤ x− Ui,j
,
otherwise,
2.12
6
Abstract and Applied Analysis
where x and x− are forward and backward difference operators in the x direction, respectively:
n−1
x Ui,j
n−1
x− Ui,j
n−1
n−1
Ui1,j
− Ui,j
hn−1
i1/2,x
n−1
n−1
Ui,j
− Ui−1,j
hn−1
i−1/2,x
,
2.13
,
and δx satisfy
hn−1
i,x n−1 n−1 δx Ui,j
≤C
Ui,j
,
2
j−L≤j≤jL i−R≤i≤iR
2.14
n−1
δx un−1
Oh,
i,j ux xi , yj , t
where L and R are positive integers independent of h. The y slope is defined analogously.
Second Step
By the Taylor expansion, for u with smooth second derivatives, we have
hni,x
Δtn
u xi1/2 , yj , tn−1/2 u ux ut O h2 Δt2 .
2
2
2.15
By 2.6, we see that
ut f − ucu · z − g1u ux − g2y − ∇ · z,
n
hi,x Δtn
Δtn
n−1/2
−
g1u ux −
g2y σ,
ui1/2,y u 2
2
2
2.16
2.17
where
σ
Δtn f − ucu · z − ∇ · z O h2 Δt2 O h2 Δt .
2
2.18
Based on 2.17 and a similar expansion about xi1 , yj , tn−1 , we define UL , UR :
L,n−1
Ui1/2,j
n−1
Ui,j
hn−1
i,x
2
R,n−1
Ui1/2,j
n−1
Ui1,j
hn−1
i,x
2
Δtn−1
Δtn−1 n−1
n−1
n−1
−
g1u Ui,j
δx Ui,j
− n−1 Γn−1
i,j1/2 − Γi,j−1/2 ,
2
2hj,y
2.19
n−1 Δtn−1
Δt
n−1
n−1
n−1
δx Ui1,j
−
g1u Ui1,j
− n−1 Γn−1
i1,j1/2 − Γi1,j−1/2 ,
2
2hj,y
Abstract and Applied Analysis
7
where
L,n−1
R,n−1
Γn−1
U
,
H
,
U
g
i,j1/2
i,j1/2
i,j1/2
L,n−1
n−1
Ui,j
Ui,j1/2
R,n−1
n−1
Ui,j1/2
Ui,j1
hn−1
j,y
2
n−1
δy Ui,j
,
hn−1
j1,y
2
2.20
n−1
δy Ui,j1
.
Third Step
n−1
With the above definitions, g n−1
1,i1/2,j , g 2,i,j1/2 is calculated as follows.
n−1
> 0, define
If c1 Ui1/2,j
L,n−1
L,n−1
L,n−1
L,n−1
g n−1
1,i1/2,j Hg1 Ui1/2,j , Ui1/2,j c1 Ui1/2,j Ui1/2,j .
2.21
n−1
If c1 Ui1/2,j
≤ 0, define
R,n−1
R,n−1
R,n−1
R,n−1
U
c
U
Ui1/2,j
g n−1
H
,
U
.
g
1
1
1,i1/2,j
i1/2,j
i1/2,j
i1/2,j
2.22
For g n−1
2,i,j1/2 , the definition is similar.
The equations 2.19–2.22 hold for elements at least one element away from the
boundary. At the left and right boundaries, we can set
L,n−1
UIR,n−1
U1/2,j
n−1 1/2,j 0,
j 1, 2, . . . , J n−1 , n 1, 2, . . . , N,
2.23
n−1
and in the slope calculation procedure, we define δx U1,j
and δx UIn−1
n−1 ,j using 2.12 with
n−1
x− U1,j
n−1
2U1,j
hn−1
1,x
,
x UIn−1
n−1 ,j
−2UIn−1
n−1 ,j
hn−1
I n−1 ,x
.
2.24
On the bottom and top boundaries, we set
n−1
Γn−1
i,1/2 Γi,J n−1 1/2 0,
in 2.19.
i 1, 2, . . . , I n−1 ,
2.25
8
Abstract and Applied Analysis
3. Projections and Lemma
We introduce three projections and a lemma to obtain error estimates. Let {
un , zn } : J →
n
n
Wh × Vh be the projection of {u, z} in mixed finite element space such that
un , ∇ · vn 0,
zn , vn − ∀vn ∈ Vhn ,
3.1
∀wn ∈ Whn .
3.2
and define πzn as the π-projection of zn such that
∇ · zn − πzn , wn 0,
These projections satisfy 21, 22:
n zn − zn Hdiv ≤ Khn ,
un − u
∂ n
∂ n
n n u − u
z − z ≤ Khn ,
∂t
∂t
Hdiv
3.3
and it is easy to see that un ∞
L ≤ K.
n
· γ at the midpoint of the boundaries is equal to
∈ Vh satisfies that the value of g
Let g
the average value of g · γ on boundaries, that is,
gn − gn , wn 0,
∇ · ∀wn ∈ Whn .
3.4
Let g
g1 , g2 . From 3.4, we can define
1
g1 xi1/2 , yj , tn n
hj,y
1
n
hj,y
yj1/2
c1 xi1/2 , yj , tn , u u xi1/2 , y, tn dy
yj−1/2
yj1/2
g1 u xi1/2 , y, tn dy.
3.5
yj−1/2
Then, g2 xi , yj1/2 , tn can be defined similarly.
Lemma 3.1. Let ξ U − u
and assume that u is sufficiently smooth. Then
n−1
n−1 ≤ K ξn−1 h Δt .
·, Un−1 − g
G
3.6
Proof. Firstly, construct u
L,n−1
,u
R,n−1
from u
given in 3.1 and 3.2 by
i1/2,j
i1/2,j
u
L,n−1
i1/2,j
u
n−1
i,j
−
Δtn 2hn−1
j,y
hn−1
i,x
2
Δtn
n−1
−
g1u u
i,j
δx u
n−1
i,j
2
Γn−1
u − Γn−1
u ,
i,j1/2 i,j−1/2 3.7
Abstract and Applied Analysis
9
at interior edges, where
n−1
δx u
i,j
⎧
⎨x u
n−1
i,j ,
n−1
n−1
if δx Ui,j
x Ui,j
,
⎩x u
n−1
− i,j ,
otherwise,
L,n−1
R,n−1
Γn−1
u
,
,
u
u
H
g
i,j1/2
i,j1/2
i,j1/2
hn−1
j,y
u
L,n−1
u
n−1
i,j i,j1/2
δy u
n−1
i,j ,
2
u
R,n−1
u
n−1
i,j1 i,j1/2
3.8
hn−1
j1,y
2
δy u
n−1
i,j1 .
And define u
R,n−1
similarly.
i1/2,j
Then, assuming g1 is twice differentiable and Lipschitz continuous, and using the
Lipschitz continuity of the Godunov flux and 2.14, it can be shown that
j2 i2 n−1
L,n−1
≤
K
L,n−1
n−1
Ul,m − u
Ui1/2,j − u
l,m .
i1/2,j
3.9
li−1 mj−1
R,n−1
A similar bound holds for |Ui1/2,j
−u
R,n−1
|.
i1/2,j
At the boundaries, we follow 2.23 and 2.25 and define
u
R,n−1
0,
u
L,n−1
1/2,j
I n−1 1/2,j
Γn−1
u
i,1/2 Γn−1
u
i,J n−1 1/2
j 1, 2, . . . , J n−1 ,
0,
i 1, 2, . . . , I
n−1
3.10
.
Define uL,n−1
, uR,n−1
analogous to u
L,n−1
,u
R,n−1
.
i1/2,j
i1/2,j
i1/2,j
i1/2,j
n−1
If c1 Ui1/2,j
> 0, we consider
g n−1
1 xi1/2 , yj , tn−1
1,i1/2,j − g
Hg1
L,n−1
L,n−1
Ui1/2,j
, Ui1/2,j
−
1
hn−1
j,y
yj1/2
yj−1/2
g1 u xi1/2 , y, tn−1 dy
10
Abstract and Applied Analysis
L,n−1
L,n−1
− Hg1 u
L,n−1
, Ui1/2,j
,u
L,n−1
Hg1 Ui1/2,j
i1/2,j
i1/2,j
L,n−1
L,n−1
L,n−1
L,n−1
−
H
u
Hg1 u
,
u
,
u
g
1
i1/2,j
i1/2,j
i1/2,j
i1/2,j
− g1 un−1/2
Hg1 uL,n−1
, uL,n−1
i1/2,j
i1/2,j
i1/2,j
⎡
⎣g1 un−1/2
−
i1/2,j
yj1/2
1
hn−1
j,y
⎤
g1 u xi1/2 , y, tn−1 dy⎦
yj−1/2
R1 R2 R3 R4 .
3.11
By the Lipschitz continuity of Hg1 and 3.9, we have
R,n−1
L,n−1
Ui1/2,j − u
−u
L,n−1
R,n−1
|R1 | ≤ K Ui1/2,j
i1/2,j i1/2,j ≤K
j2 i2 n−1
n−1
Ul,m − u
l,m .
3.12
li−1 mj−1
The similar arguments are applied to R2 to get
|R2 | ≤ K
j2 i2 n−1
ul,m − un−1
i1/2,j .
3.13
li−1 mj−1
By the consistency of Hg1 , we see
O Δt h2 .
− un−1/2
|R3 | ≤ K uL,n−1
i1/2,j
i1/2,j 3.14
|R4 | ≤ K Δt h2 .
3.15
For R4 , it is easy to have
Using 3.3, 3.12–3.15, and equivalence of norms, we have
2
n−1
n−1
n−1
n−1
g 1,i1/2,j − g1,i1/2,j
hi1/2,x hn−1
g 1 − g1n−1 ≤ K
j,y
j
i
≤ K ξn−1 Δt h .
3.16
Abstract and Applied Analysis
11
n−1
≤ 0. Defining g 2 and g2 similarly
The similar approach can be used when c1 Ui1/2,j
and following analogous arguments yields
n−1
g 2 − g2n−1 ≤ C ξn−1 Δt h .
3.17
Thus, Lemma 3.1 holds.
4. Error Estimates
At time level tn , for all wn ∈ Whn , vn ∈ Vhn , the exact solutions satisfy
un − un−1 n
,w
Δtn
∇ · zn , wn un cun · zn , wn fun , wn − ∇ · gn , wn − ρn , wn ,
z, vn un , ∇ · vn ,
zn − zn−1 n
,v
bu z au Δtn
n
z , v n
n
n
rn , vn ,
n
4.1
where ρn unt − un − un−1 /Δtn , rn aun znt − zn − zn−1 /Δtn .
Let
,
ξu U − u
βu u − u
,
ξ z Z − z,
βz z − z,
ξ z Z − πz,
4.2
βz z − πz.
Using projections 3.1 and 3.2, we subtract 2.9 from 4.1 to get
ξun − ξun−1 n
,w
Δtn
n
∇ · ξ nz , wn Un cUn · Z − un cun · zn , wn
fU − fu , w
n
n
n
ρ ,w
n
n
−G
∇· g
n
n−1
,w
n
βun − βun−1 n
,w ,
Δtn
n
ξz , vn ξun , ∇ · vn ,
n
ξ nz , vn bUn Z − bun zn , vn − rn , vn − βnz , vn
n
n−1
Z −Z
aU Δtn
n
zn − zn−1 n
− au ,v ,
Δtn
n
4.3
12
Abstract and Applied Analysis
where the last term of the first equation in 4.3 is related to the changing meshes. If the
meshes do not change, this term will be zero.
n
Taking w ξun , v ξ nz in the second equation and v ξ z in the third, then adding them
together, we have
ξun − ξun−1 n
, ξu
Δtn
⎛
n
n
bU Z − bu z
n
n
n
n
n Z
⎝
, ξ z aU −Z
Δtn
n−1
⎞
n
, ξz ⎠
−
zn − zn−1 n
au , ξz
Δtn
n
n
− Gn−1 , ξun
fUn − fun , ξun ρn , ξun ∇ · g
n
− U cU · Z − u cu · z
n
n
n
n
n
, ξun
r
n
n
, ξz
−
n
βnz , ξ z
βun − βun−1 n
, ξu .
Δtn
4.4
For 4.4, we see
I
n
n n
n n
n
bUn Z − bun zn , ξ z bUn ξz , ξ z − bUn βz , ξ z
n
bU − bu z
n
n
n
, ξz ,
4.5
II
⎛
n
n−1
⎝aUn Z − Z
Δtn
⎛
n
⎞
n
, ξz ⎠
n−1
ξ −ξ
⎝aU z nz
Δt
n
−
zn − zn−1 n
au , ξz
Δtn
n
⎞
n
, ξz ⎠
⎛
n
n−1
β −β
− ⎝aU z nz
Δt
n
⎞
n
, ξz ⎠
zn − zn−1 n
, ξz ,
aU − au Δtn
n
n
4.6
III
n
Un cUn · Z − un cun · zn , ξun
"
n
#
n
n
ξun − βun cUn · Z , ξun un cUn · ξ z − βz , ξun un cUn − cun · zn , ξun .
4.7
The last second term in 4.6 is related to the changing meshes. If the meshes do not change,
this term will be zero.
Abstract and Applied Analysis
13
Substitute 4.5–4.7 into 4.4 to get
ξun − ξun−1 n
, ξu
Δtn
n n
bUn ξ z , ξ z
⎛
n
n−1
ξ −ξ
⎝aUn z nz
Δt
⎞
n
, ξz ⎠
n
n
− Gn−1 , ξun rn , ξ z
fUn − fun , ξun ρn , ξun ∇ · g
"
n
#
n
n
n
− βnz , ξ z − ξun − βun cUn · Z , ξun − un cUn · ξ z − βz , ξun
− u cU − cu · z
n
n
n
bu − bU z
n
n
n
n
, ξun
n
, ξz
βun − βun−1 n
, ξu
Δtn
n n
bUn βz , ξ z
zn − zn−1 n
, ξz
au − aU Δtn
n
4.8
n
⎞
n
n−1
βz − βz
n
⎝aU , ξ z ⎠ T1 T2 · · · T13 .
Δtn
⎛
n
Furthermore, for the first and third terms on the left-hand side of 4.8, we have
2
1 n n
1 n
n−1 n−1
n−1 ,
ξ
,
ξ
−
ξ
−
ξ
ξ
ξ
u
u
u ,
2Δtn u u
2Δtn u
⎞
⎛
n
n−1
$
% n−1 n−1 &'
n
ξ
−
ξ
1 z
n n n
n−1
⎠
⎝aUn z
≥
−
a
U
,
ξ
,
ξ
ξz , ξz
aU
ξ
z
z z
Δtn
2Δtn
ξun − ξun−1 n
, ξu
Δtn
4.9
2
2
n−1 n
n−1 a2 a0 − ξ z ξ − ξz .
2
2Δtn z
By 4.9, we modify 4.8 as
2 n n
1 1 n n
n
ξu , ξu − ξun−1 , ξun−1 ξu − ξun−1 bUn ξ z , ξ z
n
n
2Δt
2Δt
2
$
% n−1 n−1 &'
n
n−1 a0 1
n n n
n−1
−
a
U
ξ
,
ξ
,
ξ
ξ
−
ξ
aU
ξ
z z
z
z
z 2Δtn
2Δtn z
2
n−1 a2 ≤ ξ z T1 T2 · · · T13 .
2
4.10
14
Abstract and Applied Analysis
By the Lipschitz continuity of fu, au, bu, ∂c/∂u, and 3.3, the following terms
on the right-hand side of 4.10 can be obtained:
T1 ≤ KUn − un 2 Kξun 2 ≤ Kξun 2 Kh2 ,
2 2
n ∂ u T2 ≤ KΔt 2 Kξun 2 ,
∂t 2 n−1 n 1
L t
,t ;H 2 2
n 2
n ∂ u T4 ≤ KΔt 2 K ξz ,
∂t 2 n−1 n 1
L t ,t ;H n 2
T5 ≤ K ξz Kh2 ,
T6 ≤ Kξun 2 Kh2 ,
n 2
T7 ≤ Kξun 2 K ξ z Kh2 ,
T8 ≤ KUn − un 2 Kξun 2 ≤ Kξun 2 Kh2 ,
n 2
T10 ≤ Kb1 h2 K ξz ,
n 2
T11 ≤ K ξ z Kξun 2 Kh2 ,
T12
zn − zn−1 2
n 2
2
≤ K ξ z KUn − un K Δtn 2
n 2
n 2
n −1 ∂u ≤ K ξ z Kξu KΔt Kh2 .
∂t L2 tn−1 ,tn ;H 1 4.11
n − Gn−1 in the second equation of 4.10, then we
We turn to consider T3 . Letting v g
have
n n
− Gn−1 ξun , ∇ · g
n − Gn−1 .
ξz , g
4.12
2
n n
1
n
bUn ξ z , ξ z K b0−1 g − Gn−1 2
2 2
n n
1
n
n−1
n−1 g −g
bUn ξ z , ξ z K b0−1 ≤
g − Gn−1 2
&
%
1
n−1 2
n n n
−1
2
2
bU ξ z , ξ z K b0
≤
ξu h Δt .
2
4.13
By Lemma 3.1, we have
T3 ≤
Abstract and Applied Analysis
15
Substituting 4.11 and 4.13 into 4.10, we have
2 1 1 1 n n
n
n−1 n−1
n−1 n n n
bU
,
ξ
,
ξ
−
ξ
,
ξ
−
ξ
ξ
ξ
ξ
u
u
u
u
u
u
z
z
2Δtn
2Δtn
2
2
%
$
&'
n
n−1 1
a0 n n n
n−1 n−1 n−1
ξ − ξz a ξz , ξz − a ξz , ξz
2Δtn z
2Δtn
2 %
&
n 2 n−1 n−1 2
n 2
2
2
≤ K ξ z K
ξ
Δt
h
ξ
ξu u
z 4.14
⎛
2
n ⎝ ∂u KΔt ∂t 2
⎞
∂2 u 2
⎠ T9 T13 .
2 ∂t 2 n−1 n 1
L tn−1 ,tn ;H 1 L t ,t ;H Let N be the time step at which ξun is maximum, that is,
2
N
2
ξu max ∗ ξun .
4.15
1≤n≤N
Multiplying 4.14 by 2Δtn and summing on n from 1 to N, we obtain
2
2
N N N 2 2 n
N
n−1 n
N
n−1 n n n
n
bU
Δt
−
ξ
,
ξ
a
−
ξ
a
ξ
ξ
ξu
ξu 0
0 ξ z u z z
z z
n1
n1
n1
2
N
N 2
2
0
n
0
2
2
n 2
n
ξ
≤ K Δt h K ξu Δt ξu a1 K
ξ z Δtn
z
n1
N
βun − βun−1
n1
Δtn
n1
4.16
⎛
⎞
n
n−1
N
β
−
β
n
Δtn ⎝aUn z nz , ξ z ⎠Δtn .
Δt
n1
, ξun
Assuming that the mesh is modified at most M times, and M ≤ M∗ , where M∗ is
independent of h and Δt, we get 3:
N
n1
N n n n−1 n βun − βun−1 n
n
Δt
,
ξ
βu , ξu − βu , ξu ≤ KM∗ h2 u
Δtn
n1
⎛
⎞
n
n−1
N
N 2
β
−
β
n
n
z
n
2
⎝aUn z
⎠
Δt
,
ξ
≤
Kh
ε
ξ z Δtn ,
z
n
Δt
n1
n1
where ε is a sufficiently small positive constant.
1
N 2
ξu ,
4
4.17
16
Abstract and Applied Analysis
Substituting 4.17 into 4.16, we find
2
2
N N N 2 2 n
N
n n
n−1 N
ξ z bUn ξ z , ξ z Δtn a0 ξ
−
ξ
a
ξu ξun − ξun−1 0
z z
n1
n1
n1
2
N
2
0
0
2
2
∗ 2
n 2
n
≤ K Δt h M h ξu a1 ξ z K ξu Δt
4.18
n1
K
N 2
N 2
n
n
ξ z Δtn ε ξ z Δtn n1
n1
1
N 2
ξu .
4
Using the discrete Gronwall’s lemma, we have
maxξun 2 ≤ K h2 Δt2 M∗ h2 .
n
4.19
Similarly as 4.14, we can derive
n 2
maxξ z ≤ K h2 Δt2 M∗ h2 .
n
4.20
By the triangle inequality and 3.3, we can obtain the following result.
Theorem 4.1. Assuming that the coefficients satisfy condition 1.2, the mesh is modified at most M
times, M ≤ M∗ , and u ∈ L2 H 1 , ut ∈ L2 H 1 , utt ∈ L2 H 1 , then we have
maxun − Un ≤ Kh Δt KM∗ h,
n
n
maxzn − Z ≤ Kh Δt KM∗ h,
4.21
n
where K and M∗ are positive constants independent of h and Δt.
5. Numerical Example
In this section, we give numerical experiment of the following nonlinear equations to
illustrate the efficiency of the Godunov-mixed finite element method on changing meshes:
ut ax, tuxt bx, t, uux x − cx, tux fx, t, u,
u0, t u1, t 0,
u·, 0 sin πx,
t ∈ 0, 1,
x, t ∈ 0, 1 × 0, 1,
5.1
x ∈ 0, 1,
where x, t ∈ 0, 1 × 0, 1, ax, t t2 x 0.25 0.25, bx, t, u 0.005ut2 x 0.05, cx, t 0.05tx 1 1, and fx, t, u is chosen properly so that the exact solution is u e−t sin πx.
Abstract and Applied Analysis
17
Table 1: L2 -norm estimates of Un − un .
Time
t 0.4
t 0.6
t 1.0
Case 1
0.0454
0.0253
0.0288
Case 2
0.0232
0.0135
0.0160
Case 3
0.0117
0.0064
0.0080
Case 4
0.0057
0.0032
0.0039
Case 3
0.0161
0.0164
0.0198
Case 4
0.0069
0.0080
0.0099
n
Table 2: L2 -norm estimates of Z − zn .
Time
t 0.4
t 0.6
t 1.0
Case 1
0.0537
0.0646
0.0643
Case 2
0.0307
0.0353
0.0395
Table 3: Convergence rate of L2 -norm.
n
Convergence rates of un − Un Convergence rates of zn − Z Case 1/Case 2 Case 2/Case 3 Case 3/Case 4 Case 1/Case 2 Case 2/Case 3 Case 3/Case 4
t 0.4
0.9670
0.9852
1.0392
0.8052
0.9271
1.2167
t 0.6
0.9023
1.0724
0.9974
0.8692
1.1121
1.0290
t 1.0
0.8464
1.0075
1.0242
0.7009
0.9963
1.0007
Time
In the numerical example, h and Δt change as the following four cases.
Case 1. t ∈ 0, 0.4, set h Δt 0.1; t ∈ 0.4, 0.6, set h Δt 0.05; t ∈ 0.6, 1.0, set
h Δt 0.1 and calculate 40 steps at every time interval.
Case 2. t ∈ 0, 0.4, set h Δt 0.05; t ∈ 0.4, 0.6, set h Δt 0.025; t ∈ 0.6, 1.0, set
h Δt 0.05 and calculate 80 steps at every time interval.
Case 3. t ∈ 0, 0.4, set h Δt 0.025; t ∈ 0.4, 0.6, set h Δt 0.0125; t ∈ 0.6, 1.0, set
h Δt 0.025 and calculate 160 steps at every time interval.
Case 4. t ∈ 0, 0.4, set h Δt 0.0125; t ∈ 0.4, 0.6, set h Δt 0.00625; t ∈ 0.6, 1.0, set
h Δt 0.0125 and calculate 320 steps at every time interval.
n
n
The numerical solutions Un , Z are calculated and the L2 error estimates of Un −un , Z −
zn are obtained; see Tables 1 and 2.
n
In Table 3, convergence rates of Un − un , Z − zn are given. The figures of convergence
rate are shown by Figure 1. The convergence rate of Un − un in Figure 1a is one order,
n
and in Figure 1b the convergence rate of Z − zn is a little smaller than one order at the
n
beginning. But the convergence rate Z − zn will be close to one order when h minishes,
which is consistent with the analysis in this paper.
For Case 4, we compare the exact solution {u, z} with the approximate solution {U, Z}
at time t 0.25, 0.5, and 0.75, respectively; see Figure 2. From Figures 2a and 2b, we can
see that the approximate solutions are very close to the exact solutions.
Abstract and Applied Analysis
−3.5
−2.5
−4
−3
−4.5
−3.5
log e
log e
18
−5
−5.5
−6
−4.5
−4
−4.5
−4
−3.5
−3
−2.5
−5
−4.5
−2
−4
−3.5
−3
−2.5
−2
− log h
− log h
a Convergence rates of u
b Convergence rates of z
Figure 1: Convergence rate figures.
2.5
0.8
0.7
t = 0.25
2
t = 0.25
1.5
0.6
0.5
−ux
U 0.4
t = 0.75
1
t = 0.5
0.5
t = 0.5
t = 0.75
0.3
0
−0.5
−1
0.2
−1.5
0.1
−2
0
−2.5
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
Numerical
Exact
a Comparing u with U
1
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
1
Numerical
Exact
b Comparing z with Z
Figure 2: Compare figures for Case 4 at time t 0.25, 0.5, and 0.75.
Acknowledgments
The author would like to thank the referees for their constructive comments leading to
an improved presentation of this paper. This work was supported by the NSF of China
no. 11271231, the NSF of Shandong Province no. ZR2010AQ019, the Scientific Research
Award Fund for Excellent Middle-Aged and Young Scientists of Shandong Province no.
BS2009NJ003, and Science and Technology Development Planning Project of Shandong
Province no. 2011GGH20118, China.
Abstract and Applied Analysis
19
References
1 K. Miller and R. N. Miller, “Moving finite elements. I,” SIAM Journal on Numerical Analysis, vol. 18,
no. 6, pp. 1019–1032, 1981.
2 K. Miller, “Moving finite elements. II,” SIAM Journal on Numerical Analysis, vol. 18, no. 6, pp. 1033–
1057, 1981.
3 G. P. Liang and Z. M. Chen, “A full-discretization moving FEM with optimal convergence rate,”
Chinese Journal of Numerical Mathematics and Applications, vol. 12, no. 4, pp. 91–111, 1990.
4 D. Q. Yang, “The mixed finite-element methods with moving grids for parabolic problems,”
Mathematica Numerica Sinica, vol. 10, no. 3, pp. 266–271, 1988.
5 H. L. Song and Y. R. Yuan, “An upwind-mixed method on changing meshes for two-phase miscible
flow in porous media,” Applied Numerical Mathematics, vol. 58, no. 6, pp. 815–826, 2008.
6 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.
7 D. M. Shi, “The initial-boundary value problem for a nonlinear equation of migration of moisture in
soil,” Acta Mathematicae Applicatae Sinica, vol. 13, no. 1, pp. 31–38, 1990.
8 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.
9 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.
10 J. Zhu, “Finite element methods for nonlinear Sobolev equations,” Northeastern Mathematical Journal,
vol. 5, no. 2, pp. 179–196, 1989.
11 T. J. Sun and K. Y. Ma, “Finite difference-streamline diffusion method for quasi-linear Sobolev
equations,” Numerical Mathematics, vol. 23, no. 4, pp. 340–356, 2001.
12 T. J. Sun and D. P. 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.
13 J. S. Zhang, D. P. Yang, and J. Zhu, “Two new least-squares mixed finite element procedures for
convection-dominated Sobolev equations,” Applied Mathematics, vol. 26, no. 4, pp. 401–411, 2011.
14 C. Chavent and J. Jaffer, Mathematical Models and Finite Elements for Reservoir Simulation, North
Holland, Amsterdam, The Netherlands, 1996.
15 J. B. Bell, C. N. Dawson, and G. R. Shubin, “An unsplit, higher order godunov method for scalar
conservation laws in multiple dimensions,” Journal of Computational Physics, vol. 74, no. 1, pp. 1–24,
1988.
16 C. N. Dawson, “Godunov-mixed methods for immiscible displacement,” International Journal for
Numerical Methods in Fluids, vol. 11, no. 6, pp. 835–847, 1990.
17 J. B. Bell, P. Colella, and H. M. Glaz, “A second-order projection method for the incompressible
Navier-Stokes equations,” Journal of Computational Physics, vol. 85, no. 2, pp. 257–283, 1989.
18 C. N. Dawson, “Godunov-mixed methods for advective flow problems in one space dimension,”
SIAM Journal on Numerical Analysis, vol. 28, no. 5, pp. 1282–1309, 1991.
19 C. Dawson, “Godunov-mixed methods for advection-diffusion equations in multidimensions,” SIAM
Journal on Numerical Analysis, vol. 30, no. 5, pp. 1315–1332, 1993.
20 T. Arbogast, M. F. Wheeler, and I. Yotov, “Mixed finite elements for elliptic problems with tensor
coefficients as cell-centered finite differences,” SIAM Journal on Numerical Analysis, vol. 34, no. 2, pp.
828–852, 1997.
21 P. A. Raviart and J. M. Thomas, “A mixed finite element method for 2nd order elliptic problems,” in
Mathematical Aspects of the Finite Element Methods, vol. 606 of Lecture Notes in Mathematics, pp. 292–315,
Springer, Berlin, Germany, 1977.
22 J. Douglas, Jr. and J. E. Roberts, “Global estimates for mixed methods for second order elliptic
equations,” Mathematics of Computation, vol. 44, no. 169, pp. 39–52, 1985.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014