A SPLIT LEAST-SQUARES CHARACTERISTIC MIXED FINITE ELEMENT METHOD FOR THE CONVECTION DOMINATED SOBOLEV EQUATIONS†

Journal of Applied Mathematics & Informatics.
2016.
Jan,
34(1_2):
19-34

- Received : August 08, 2015
- Accepted : October 16, 2015
- Published : January 30, 2016

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In this paper, we present a split least-squares characteristic mixed finite element method(MFEM) to get the approximate solutions of the convection dominated Sobolev equations. First, to manage both convection term and time derivative term efficiently, we apply a least-squares characteristic MFEM to get the system of equations in the primal unknown and the flux unknown. Then, we obtain a split least-squares characteristic MFEM to convert the coupled system in two unknowns derived from the least-squares characteristic MFEM into two uncoupled systems in the unknowns. We theoretically prove that the approximations constructed by the split least-squares characteristic MFEM converge with the optimal order in
L
^{2}
and
H
^{1}
normed spaces for the primal unknown and with the optimal order in
L
^{2}
normed space for the flux unknown. And we provide some numerical results to confirm the validity of our theoretical results.
AMS Mathematics Subject Classification : 65M15, 65N30.
where Ω is a bounded convex domain in ℝ
^{m}
with 1 ≤
m
≤ 3 with boundary ∂Ω,
c
(
x
),
d
(
x
),
a
(
x
),
b
(
x
),
f
(
x , t
), and
u
_{0}
(
x
) are given functions. The Sobolev equation which represents the flow of fluids through fissured rock, the migration of the moisture in soil, the physical phenomena of thermodynamics and other applications as described in
[2
,
19
,
20]
, is one of most principal partial differential equations. For the existence and uniqueness results of the solutions of the equation (1.1), refer to
[8]
.
For the problems with no convection term, mixed finite element methods
[11
,
16
,
18
,
22]
, least-squares methods
[12
,
18
,
21
,
22]
, and discontinuous Galerkin methods
[14
,
15]
were used for numerical treatments. In the case that a conventional (least-squares) MFEM is applied, we generally needs to solve the coupled system of equations in two unknowns, which brings to difficulties in some extent. So, in
[18]
, a split least-squares mixed finite element method for reaction-diffusion problems was firstly introduced to solve the uncoupled systems of equations in the unknowns.
For the partial differential equations with a convection term, a characteristic (mixed) finite element method is one of the useful methods
[1
,
3
,
4
,
5
,
6
,
7
,
10
,
13]
because it reflects well the physical character of a convection term and also it treats efficiently both convection term and time derivative term. Gao and Rui
[9]
introduced a split least-squares characteristic MFEM to approximate the primal unknown
u
and the flux unknown −
a
∇
u
of the equation (1.1) and obtained the optimal convergence in
L
^{2}
(Ω) norm for the primal unknown and in
H
(
div
, Ω) norm for the flux unknown. And Zhang and Guo
[23]
introduced a split least-squares characteristic mixed element method for nonlinear nonstationary convection-diffusion problem to approximate the primal unknown and the flux unknown and obtained the optimal convergence in
L
^{2}
(Ω) norm for the primal unknown and in
H
(
div
, Ω) norm for the flux unknown.
In this paper, we apply a split least-squares characteristic characteristic mixed finite element method (MFEM) to achieve two uncoupled system of equations, one of which is for approximations to the primal unknown
u
and the other of which is for ones to the flux unknown
σ
= −(
a
(
x
)∇
u_{t}
+
b
(
x
)∇
u
) of the equation (1.1). And we analyze the optimal order of convergence in
L
^{2}
and
H
^{1}
normed spaces for the approximations. In section 2, we introduce necessary assumptions and notations, and in section 3, we construct finite element spaces on which we compose the approximations of two unknowns. In section 4, by adopting a split least-squares characteristic MFEM, we construct the approximations of the primal unknown and the unknown flux and establish the convergence of optimal order in
L
^{2}
and
H
^{1}
normed spaces for the primal unknown and the convergence of optimal order in
L
^{2}
normed space for the flux unknown. In section 5, we provide some numerical results to confirm the validity of the theoretical results obtained in section 4.
s
≥ 0 and 1 ≤
p
≤ ∞, we denoted by
W^{s,p}
(Ω) the Sobolev space endowed with the norm
where
k
= (
k
_{1}
,
k
_{2}
, · · · ,
k_{m}
), |
k
| =
k
_{1}
+
k
_{2}
+· · ·+
k_{m}
,
and
k_{i}
is a nonnegative integer, for each
i
, 1 ≤
i
≤
m
. If
p
= 2, we simply denote
H^{s}
(Ω) =
W
^{s,2}
(Ω) and ║
ϕ
║
_{s}
= ║
ϕ
║
_{s,2}
. And also in case that
s
= 0, we simply write ║
ϕ
║. We let
H ^{s}
(Ω) = {
u
= (
u
_{1}
,
u
_{2}
, · · · ,
u_{m}
) |
u_{i}
∈
H^{s}
(Ω), 1 ≤
i
≤
m
} with the norm
and
W
=
H
(
div
, Ω).
If
ϕ
(
x, t
) belongs to a Sobolev space equipped with a norm ║·║
_{X}
for each
t
, then we let
In case that
t
_{0}
=
T
, we denote
L^{p}
(0,
T
:
X
) and
L
^{∞}
(0,
T
:
X
) by
L^{p}
(
X
) and
L
^{∞}
(
X
), respectively. Let
H^{q,∞}
(
X
) = {
ϕ
(
x, t
) |
ϕ
(
x, t
),
ϕ_{t}
(
x, t
), · · · ,
ϕ_{q}
(
x, t
) ∈
L
^{∞}
(
X
)} for a nonnegative integer
q
.
We consider the problem (1.1) with the coefficients satisfying the following assumption:
(A).
There exist
c
_{∗}
,
c
^{∗}
,
d
^{∗}
,
a
_{∗}
,
a
^{∗}
,
b
_{∗}
, and
b
_{∗}
such that 0 <
c
_{∗}
<
c
(
x
) ≤
c
^{∗}
, 0 < |
d
(
x
)| ≤
d
^{∗}
, 0 <
a
_{∗}
<
a
(
x
) ≤
a
^{∗}
, and 0 <
b
_{∗}
<
b
(
x
) ≤
b
^{∗}
, for all
x
∈ Ω, where
ℇ_{h}
= {
E
_{1}
,
E
_{2}
, · · · ,
E_{Nh}
} be a family of regular finite element subdivision of Ω. We let
h
denote the maximum of the diameters of the elements of
ℇ_{h}
. If
m
= 2, then
E_{i}
is a triangle or a quadrilateral, and if
m
= 3, then
E_{i}
is a 3-simplex or 3-rectangle. Boundary elements are allowed to have a curvilinear edge (or a curved surface).
We denote by
V_{h}
×
W _{h}
the Raviart-Thomas-Nedlec space associated with
ℇ_{h}
. For each triangle (or 3-simplex) element
E
∈
ℇ_{h}
, we define
V_{h}
(
E
) =
P_{k}
(
E
), and
W _{h}
(
E
) =
P_{k}
(
E
)
^{m}
⊕ (
x
_{1}
,
x
_{2}
, · · · ,
x
_{m}
)
^{T}
P_{k}
(
E
) where
P_{k}
(
E
) is the set of polynomials of total defree ≤
k
difined on
E
. Now we define the finite element spaces
And also in case that
E
is a rectangle (or a parallelogram), we adopt analogous modification to construct
V_{h}
and
W _{h}
.
Let
P_{h}
×
∏
_{h}
:
V
×
W
→
V_{h}
×
W _{h}
denote the Raviart-Thomas
[17]
projection which satisfies
Then, obviously, (∇ ·
w
,
v
−
P_{h}v
) = 0 holds for each
v
∈
V
and each
w
∈
W _{h}
and div
∏
_{h}
=
P_{h}
div is a function from
W
onto
V_{h}
. It is proved that the following approximation properties hold
[17]
:
and
ν
=
ν
(
x , t
) be the unit vector in the direction of (
d
(
x
),
c
(
x
)). Then, we have
Hence the problem (1.1) can be written in the form
By introducing the flux term
σ
= −(
a
(
x
)∇
u_{t}
+
b
(
x
)∇
u
), the problem (4.1) can be rewritten as follows:
For a positive integer
N
, let Δ
t
=
T/N
and
t^{n}
=
n
Δ
t
,
n
= 0, 1, · · · ,
N
. Choosing
t
=
t^{n}
in (4.2) and discretizing it with respect to
t
by applying the backward Euler method along
ν
-characteristic tangent at (
x , t^{n}
), we get
where
. Therefore we have
where
Now let
ã
(
x
) =
a
(
x
) +
b
(
x
)△
t
. By multiplying the first equation of (4.3) by
and the second equation by
, we have the equivalent system of equations
For (
v, τ
) ∈
V
×
W
, we define a least-squares functional
J
(
v, τ
) as follows
Then the least-squares minimization problem is to find
s
solution (
u^{n}
,
σ ^{n}
) ∈
V
×
W
such that
If we define the bilinear form
A
on (
V
×
W
)
^{2}
by
then the weak formulation of the minimization problem becomes as follows: find (
u^{n}
,
σ ^{n}
) ∈
V
×
W
such that
Based on (4.6), we derive the following least-squares characteristic MFEM scheme: find
∈
V_{h}
×
W _{h}
satisfying
Lemma 4.1.
For (
v, τ
) ∈
V
×
W
, we have
Proof
. From the definition of the bilinear form (4.5), we have
Letting
v_{h}
= 0 in (4.7) and applying the definition of the bilinear form
A
, we have
which implies that
Since
, we have
Letting
τ _{h}
= 0 in (4.7) and applying the definition of the bilinear form
A
, we have
Finally, we derive a split least-squares characteristic MFEM: find
∈
V_{h}
×
W _{h}
satisfying:
For the error analysis, we define an elliptic projection
ũ
(
x, t
) of
u
(
x, t
) onto
V_{h}
satisfying
Obviously, by the assumption (A), there exists a unique elliptic projection
ũ
(
x, t
) ∈
V_{h}
. Now we let
η
=
u
−
ũ
and
ξ
=
u_{h}
−
ũ
so that
u
−
u_{h}
=
η
−
ξ
.
Hereafter a constant
K
denotes a generic positive constant depending on Ω and
u
, but independent of
h
and Δ
t
, and also any two
Ks
in different places don’t need to be the same. We state the error bounds of
η
below, the proofs of which can be found in
[14
,
15]
.
Theorem 4.2
(
[14]
).
If u_{t}
∈
L
^{2}
(
H^{s}
(Ω))
and u
_{0}
∈
H^{s}
(Ω),
then there exists a constant K, independent of h, such that
(i) ║
η
║ +
h
║
η
║
_{1}
≤
Kh^{μ}
(║
u_{t}
║
_{L}
_{2}
_{(Hs)}
+ ║
u
_{0}
║
_{s}
),
(ii) ║
η
║ +
h
║
η_{t}
║
_{1}
≤
Kh^{μ}
(║
u_{t}
║
_{L}
_{2}
_{(Hs)}
+ ║
u
_{0}
║
_{s}
),
where
μ
= min(
k
+ 1,
s
).
Theorem 4.3
(
[15]
).
If u_{t}
∈
L
^{2}
(
H^{s}
(Ω))
, u_{tt}
(
t
) ∈
H^{s}
(Ω),
and u
_{0}
∈
H^{s}
(Ω)
, then there exists a constant K, independent of h, such that
where
μ
= min(
k
+ 1,
s
).
Lemma 4.4.
If u
∈
H
^{1,∞}
(
H
^{2}
(Ω))
and u_{tt}
(
t
) ∈
L
^{2}
(Ω)
, then
Proof
. By applying Taylor’s expansion, we obviously have the estimations of
. □
Theorem 4.5.
In addition to the hypotheses of Theorem 4.2 and 4.3, if u
(
t
) ∈
H^{s}
(Ω)
, u
∈
H
^{1,∞}
(
H
^{2}
(Ω))
, and
Δ
t
=
O
(
h
)
, then
where μ
= min(
k
+ 1,
s
).
Proof
. Subtracting (4.1) at
t
=
t^{n}
from (4.8), we get the equation
Now we set
in (4.11). Then letting three terms of the left-hand side of (4.11) by
L
_{1}
,
L
_{2}
, and
L
_{3}
, respectively, we get the following estimates for
L
_{1}
,
L
_{2}
, and
L
_{3}
Now let
ϵ
> 0 be sufficiently small, but independent of
h
and Δ
t
. Since
for some
,
R
_{1}
can be estimated as follows:
By noting that
for some
, we can estimate
R
_{2}
as follows:
By Lemma 4.4, we obviously get
By (4.10), Theorem 4.3, and the Taylor expansion, we have
where
. By the Taylor expansion, we get
for some
. Now by applying the bounds of
L
_{1}
∼
L
_{3}
and
R
_{1}
∼
R
_{6}
to (4.11), we obtain
which yields that for sufficiently small
ϵ
> 0
Now we sum up both sides of (4.12) from
n
= 1 to
n
=
N
to get
By the discrete-type Gronwall inequality, we get
from which we get by Poincare’s inequality
Therefore, by using Theorem 4.2 and the triangular inequality, we obtain
By applying Lemma 4.1 to (4.6), we get
and hence, letting
v
= 0, we obtain
And letting
v_{h}
= 0 in (4.7) and applying the definition of the bilinear form
A
, we get
which implies that
Therefore we have
For
σ
∈
W
, we define an elliptic projection
∈
W _{h}
of
σ
satisfying
where
λ
is a positive real n
u_{m}
ber. By applying the Lax-Milgram lemma, the existence of
can be obtained.
Lemma 4.6.
If σ
∈
W
∩
H ^{s}
(Ω)
, then there exists a constant K
> 0
such that
where μ
= min(
k
+ 1,
s
).
Proof
. By the difinition of
and (3.5), we get
and so
Therefore, by (3.4), we have
for sufficiently small
λ
> 0. We let
φ
∈
H
^{2}
(Ω) be the solution of an elliptic problem
where
n
denotes the outward normal unit vector to ∂Ω. By the regularity property of the elliptic problem, we have ║
φ
║
_{2}
≤
K
║
σ
−
∥. Using (3.4), (3.5), (4.15), (4.16), and (4.17), we obtain the following estimation
Now if we choose
h
sufficiently small, then we get ║
σ
−
║ ≤
Kh^{μ}
║
σ
║
_{s}
. □
Theorem 4.7.
In addition to the hypotheses of Theorem 4.5, if σ
∈
W
∩
H ^{s}
(Ω)
, then
where μ
= min(
k
+ 1,
s
).
Proof
. By subtracting (4.14) from (4.13), we have
Now we let
π
=
σ
−
,
ρ
=
−
σ _{h}
. From (4.18), we get
Choosing
τ _{h}
=
ρ ^{n}
in (4.19) and applying the integration by parts, we obtain
Note that
and
By applying (4.15) to (4.20), we get
By using Lemma 4.4, Lemma 4.6, and Theorem 4.5, we get
Let
ψ^{n}
∈
H
^{2}
(Ω) be the solution of an elliptic problem
where
n
denotes the outward normal unit vector to ∂Ω. By the regularity property of the elliptic problem, we have ║
ψ^{n}
║
_{2}
≤
K
║
ρ ^{n}
║. We let
be the elliptic projection of
ψ^{n}
onto
W _{h}
defined by exactly the same way as (4.15). Then using (4.19) and (4.22) with
τ _{h}
=
, we get
By using (4.21), Lemma 4.6, and the fact that ║
ψ ^{n}
−
║ ≤
ch
^{2}
║
ψ ^{n}
║
_{2}
, we get the estimations of
I
_{1}
∼
I
_{3}
as follows:
By the definitions of
and Theorem 4.5, we have the estimations of
I
_{4}
∼
I
_{6}
as follows:
Using the definition
ψ^{n}
, Theorem 4.2, and Lemma 4.4, we estimate
I
_{7}
∼
I
_{9}
as follows:
By applying the estimations of
I
_{1}
∼
I
_{9}
to (4.23), we obtain
Therefore ║
ρ ^{n}
║ ≤
K
(
h^{μ}
+ Δ
t
) holds for sufficiently small
h
> 0. Thus by the triangular inequality and Lemma 4.6, we obtain the result of this theorem. □
c
(
x
) =
d
(
x
) = 1,
a
(
x
) =
b
(
x
) = 0.001 and Ω = [0, 1].
We construct the approximation of
u
(
x, t
) on the finite element space consisting of the piecewise linear polynomials defined on the uniform grids and the approximation of
σ
(
x, t
) on the finite element space consisting of the piecewise quadratic polynomials defined on the uniform grids. Choose the exact solution
u
(
x, t
) as follows:
and compute
f
(
x, t
) =
u_{t}
+
u_{x}
− 10
^{−3}
_{uxx}
− 10
^{−3}
_{utxx}
by substituting
u
(
x, t
) defined in (5.1). Notice that
u
(
x, t
) ∈
H
^{4}
(Ω) and
σ
(
x, t
) ∈
H
^{2}
(Ω)
The numerical results for
u_{h}
(
x
) at
T
= 0.4 are given in
Table 1
in terms of the space mesh size
h
and the time mesh size △
t
. We know from
Table 1
that the convergence orders in
L
^{2}
and
H
^{1}
norms for
u_{h}
at
T
= 0.4 are consistent with the results in Theorem 4.5.
The estimates for u_{h}
The corresponding numerical results for
σ_{h}
at
T
= 0.4 are given in
Table 2
in terms of the space mesh size
h
and the time mesh size △
t
. We know from
Table 2
that the convergence order in
L
^{2}
norm for
σ_{h}
at
T
= 0.4 is consistent with the result in Theorem 4.7.
The estimates for σ_{h}
Mi Ray Ohm received her BS degree from Busan National University and Ph.D. from Busan National University under the direction of Professor Ki Sik Ha. She is currently a professor at Dongseo University. Her research interest is numerical analysis on partial differential equations.
Division of Mechatronics Engineering, Dongseo University, Busan 47011, Korea.
e-mail: mrohm@dongseo.ac.kr
Jun Yong Shin received his BS degree from Busan National University and Ph.D. degree from University of Texas at Arlington under the direction of Professor R. Kannan. He is currently a professor at Pukyong National University. His research interest is numerical analysis on partial differential equations.
Department of Applied Mathematics, College of Natural Sciences, Pukyong National University, Busan 48513, Korea.
e-mail: jyshin@pknu.ac.kr

Convection dominated Sobolev equations
;
A split least-squares method
;
characteristic mixed finite element method
;
convergence of optimal order

1. Introduction

In this paper we consider the following convection dominated Sobolev equation:
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2. Assumption and notations

For an
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3. Finite element spaces

Before preceding the numerical scheme, we let
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4. OptimalL2error analysis

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5. Numerical example

In this section, we will present some numerical results to verify the convergence order of the split least-squares CMFEM proposed in (4.8) and (4.9). For the sake of convenience, we consider the one dimensional convection dominated Sobolev equation (1.1) with
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The estimates foruh

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The estimates forσh

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Citing 'A SPLIT LEAST-SQUARES CHARACTERISTIC MIXED FINITE ELEMENT METHOD FOR THE CONVECTION DOMINATED SOBOLEV EQUATIONS†
'

@article{ E1MCA9_2016_v34n1_2_19}
,title={A SPLIT LEAST-SQUARES CHARACTERISTIC MIXED FINITE ELEMENT METHOD FOR THE CONVECTION DOMINATED SOBOLEV EQUATIONS†}
,volume={1_2}
, url={http://dx.doi.org/10.14317/jami.2016.019}, DOI={10.14317/jami.2016.019}
, number= {1_2}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={OHM, MI RAY
and
SHIN, JUN YONG}
, year={2016}
, month={Jan}