In this paper we consider the nonlinear parabolic problems with mixed boundary condition. Under comparatively mild conditions of the coefficients related to the problem, we construct the discontinuous Galerkin approximation of the solution to the nonlinear parabolic problem. We discretize spatial variables and construct the finite element spaces consisting of discontinuous piecewise polynomials of which the semidiscrete approximations are composed. We present the proof of the convergence of the semidiscrete approximations in
L
^{∞}
(
H
^{1}
) and
L
^{∞}
(
H
^{2}
) normed spaces.
AMS Mathematics Subject Classification : 65M15, 65N30.
1. Introduction
Discontinuous Galerkin (DG) finite element methods employ discontinuous piecewise polynomials to approximate the solutions of differential equations and impose interelement continuity weakly. Even though DG methods often have been involved with large number of degrees of freedom than the classical Galerkin method, DG methods are adopted widely in a variety of differential equations. DG methods were introduced for the numerical solutions of firstorder hyperbolic system, but independently they are proposed as nonconforming schemes for the numerical solutions of 2nd order elliptic problems by Nitsche
[10]
. Recently there has been renewed interest in DG methods due to their efficient properties which include a high degree of locality, the flexibility of locally varying the degree of polynomial in adaptive hp version approximations since no continuity requirement is imposed.
Much attention have been devoted to the analysis of DG methods applied to elliptic problem
[6
,
7
,
8]
as well as to several other types of nonlinear equations including timedependent convectiondiffusion equations
[3]
, nonFickian diffusion equation
[14]
, CamassaHolm equation
[18]
, solid viscoelasticity problems
[15]
, Maxwell equations
[4]
, Navier Stokes equations
[16]
, KellerSegel chemotaxis model
[5]
and reactive transport problem
[17]
.
In this paper we consider the DG methods applied to parabolic problems. In
[13]
Rieviere and Wheeler initiated to adopt DG method and develop DG approximations to parabolic problems. They constructed discontinuous and time discretized approximations and obtained the optimal convergence order of spatial error in
H
^{1}
and time truncation error in
L
^{2}
normed space. In
[11]
the authors applied DG method to parabolic problem with homogeneous Neumann boundary condition and constructed DG spatial discretized approximations using a penalty term and obtained an optimal
L
^{∞}
(
L
^{2}
) error estimate. In addition the authors
[12]
applied DG method to construct the fully discrete approximations for the parabolic problems with homogeneous Neumann boundary condition and obtained the optimal order of convergence in
ℓ
^{∞}
(
L
^{2}
) normed space.
And also Lasis and Süli
[9]
considered the hpversion DG method with interior penalty for semilinear parabolic equations to construct spatial discretized approximations and obtained an optimal
L
^{∞}
(
H
^{1}
) and
L
^{∞}
(
L
^{2}
) error estimates.
In this paper we consider the semidiscrete DG approximations of the nonlinear parabolic equations. Compared to the previous works in this paper we require very weak conditions on the terms characterizing the nonlinearity of the parabolic problem. In this paper we weaken the conditions of the tensor coefficient and the forcing term so that they are assumed to be locally Lipschitz continuous only. In addition, the parabolic problem considered in this paper is related with mixed nonhomogeneous Dirichletnonhomogeneous Neumann boundary conditon so that we manage the most generalized boundary condition. The rest of this paper is as follows. In Section 2 we introduce our parabolic problem to be considered and some notations and we construct finite element space. In Section 3, we develop some auxiliary projection onto finite element space and we prove its convergence of optimal order. In Section 4 we construct the semidiscrete approximation and prove its existence and finally we provide the error analysis of the semidiscrete approximations.
2. The problem and notations
Consider the following nonlinear parabolic differential equation:
where Ω is a bounded open convex domain in ℝ
^{d}
, 1 ≤
d
≤ 3,
∂
Ω is the boundary of Ω,
∂
Ω
_{N}
∪
∂
Ω
_{D}
=
∂
Ω,
∂
Ω
_{N}
∩
∂
Ω
_{D}
=
ϕ
and
n
is a unit outward normal vector to
∂
Ω.
Assume that
(A1).
a
(
x, u
(
x, t
)) is continuous at
(A2). There exists a positive constant
a
_{*}
such that
a
(
x, u
(
x, t
)) ≥
a
_{*}
,
Let
be a subdivision of Ω, where
E_{i}
is an interval if
d
= 1, and in case of
d
= 2(
d
= 3)
E_{i}
is a triangle or a quadrilateral (a symplex or parallelogram) which may have one curved edge (face). Let
h_{i}
be the diameter of
E_{i}
and
h
= max{
h_{i}
: 1 ≤
i
≤
N_{h}
}. We assume that there exists a constants
δ
such that
δ
^{−1}
h
≤
h^{i}
≤
δh
, 1 ≤
i
≤
N_{h}
.
Let
ε_{h}
be the set of the edge of
E_{i}
, 1 ≤
i
≤
N_{h}
and we let
where
is (
d
−1) dimensional measure defined in ℝ
^{d−1}
. If
e
=
∂E_{i}
∩
∂E_{j}
with
i
<
j
, the unit outward normal vector
n_{i}
to
E_{i}
is taken as the unit vector
n
associated with
e
The
L
^{2}
inner product is denoted by (·, ·) and we denote usual
L
^{2}
norm defined on
E
by ∥ · ∥
_{E}
, and usual
L
^{∞}
norm by ∥ · ∥
_{∞,E}
. In both cases we may skip
E
if
E
= Ω. Let
H^{s}
(
E
) be the Sobolev space equipped with the usual Sobolev norm
where
If
E
= Ω, we simply denote it by ∥·∥
_{s}
and if
s
= 0 denote it by ∥·∥
_{E}
. We denote the usual seminorm defined on
E
by  · 
_{s,E}
. And also we denote
W
^{s,∞}
(
E
) = {
v

D^{l}v
∈
L
^{∞}
, ∀ 
l
 ≤
s
} equipped with the norm
If
E
= Ω then for our convevience we skip
E
in the notation of
W
^{s,∞}
(
E
). Now we let
H^{s}
(Ω
_{h}
) = {
v

v

_{Ei}
∈
H^{s}
(
E_{i}
) 1 ≤
i
≤
N_{h}
}. If
v
∈
H^{s}
(Ω
_{h}
) with
we define the average {
v
} and the jump ⟨
v
⟩ functions as follows: For
e
∈
∂E_{i}
∩
∂E_{j}
with
i
<
j
then
For
e
∈
∂
Ω
_{D}
,
Now we define the following broken norm on
H
^{2}
(Ω
_{h}
):
To continue our analysis we may assume that
E_{i}
is a triangle. For the case that
E_{i}
is a rectangle we may develop the analogous theories. We let
be the space of piecewise polynomials defined as
where
P_{r}
(
E_{j}
) is the set of polynomials of total degree ≤
r
.
3. Approximation properties and elliptic projection
Hereafter
C
denotes a generic positive constant independent of
h
and any two
C s
in different positions don’t need to be the same. The following approximation properties are proved in
[1
,
2]
.
Lemma 3.1.
Let E
∈ Ω
_{h}
,
e be an edge of E and v
∈
H^{s}
(
E
).
Then there exist a positive constant independent of v, r and h and a sequence
r
= 1, 2, · · · ,
such that for any
0 ≤
q
≤
s
,
hold where μ
= min(
r
+ 1,
s
).
Moreover if e
=
∂E_{i}
∩
∂E_{j}
then
holds
.
Lemma 3.2.
Let E
∈ Ω
_{h}
,
e be an edge of E and n be a normal vector associated with e. Then there exists a positive constant C such that
∀
v
∈
H
^{1}
(
E
)
Now we let
be the interpolation of
u
satisfying the approximation properties of Lemma 3.1. By applying Lemma 3.1 we obviously have the following Lemma.
Lemma 3.3.
If u
∈
H^{s}
(Ω),
then
satisfies the following approximation property
where μ
= min(
r
+ 1,
s
).
Proof
. By applying Lemma 3.1 and 3.2, we get
We define the following bilinear form
A^{β}
(
a, u
:
v,w
) on
For a
α
> 0 we let
Lemma 3.4.
For any v,w
∈
H^{s}
(Ω
_{h}
)
with s
≥ 2
holds.
Proof
. For any
v,w
∈
H^{s}
(Ω
_{h}
) with
s
≥ 2,
Lemma 3.5.
If β is sufficiently large, then there is a constant
such that
Proof
. For
and
δ
> 0
holds. By applying Lemma 3.2 we get for sufficiently large
β
and
δ
,
since
By applying Lemmas 3.4 and 3.5 there exists
satisfying
By LaxMilgram Lemma,
ũ
satisfies
Now we let
Lemma 3.6.
Let G be a linear mappping defined on H
^{2}
(Ω
_{h}
)
and suppose that there exists w
∈
H
^{2}
(Ω
_{h}
)
satisfying
Suppose that there exist positive constants
K
_{1}
and
K
_{2}
such that
where
Then we have the following estimation
Proof
. Let
ϕ
be the solution of
Then by the regularity property of elliptic problem we get ∥
ϕ
∥
_{2}
≤
C
∥
w
∥. By Lemma 3.3 there exists
an interpolation
ϕ
satisfying
By (3.3) we have
which implies that
Theorem 3.1.
Suppose that u
(·,
t
) ∈
H^{s}
,
u_{t}
(·,
t
) ∈
H^{s}
then the following error estimations hold:
where μ
= min(
r
+1,
s
).
And also if
are bounded for e
∈
ε_{h}
.
Proof
. By Lemma 3.4 and 3.5, we have
so that
By (3.1), (3.2) and Lemma 3.6 with
G
(
v
) = 0, we get
Now we differentiate
with respect to
t
to obtain
where
By the similar process as the proof of Lemma 3.4 we have with
v
∈
H
^{2}
(Ω
_{h}
),
and with
Therefore, by Lemma 3.6
By Lemma 3.5, Lemma 3.4 and (3.6) we have
which implies that by Lemma 3.3,
Hence we get
Now we substitute (3.8), (3.2) and (3.5) into (3.7) to get
If
then
holds. Now we let
e
=
E_{i}
∩
E_{j}
. By Lemma 3.1
holds.
4. Spatial discretized approximation and error analysis.
The discontinuous Galerkin method of the problem (1.1) reads as follows: find
such that
where
and
P_{h}
(
u
_{0}
(
x
)) denotes the approximation of
u
_{0}
(
x
) generated by Lemma 3.1. From (2.1)
u
(
x, t
) satisfies
Theorem 4.1.
There exists u_{h}
(
x, t
)
satisfying
(4.1).
If f
(
x, t, u
)
and a
(
x, u
)
are locally Lipschitz continuous in t and u then there exists a unique u_{h}
(
x, t
)
locally. And also if f
(
x, t, u
)
and a
(
x, u
)
are globally Lipschitz continuous in t and u then the unique existence holds globally
.
Proof
. Let
be a basis of
and suppose that
From (4.1) we have
Let
α
(
t
) = (
α
_{1}
(
t
),
α
_{2}
(
t
), · · · ,
α_{m}
(
t
))
^{T}
. (4.3) can be represented as the following system
where
M
= (
M_{ij}
)
_{1≤i,j≤m}
,
N
(
α
(
t
)) = (
N_{ij}
(
α
))
_{1≤i,j≤m}
are symmetric matrices and
F
(
α
(
t
)) =
F_{j}
(
α
(
t
))
_{1≤j≤m}
and
L
(
t
) = (
L_{j}
(
t
))
_{1≤j≤m}
are vectors.
M,N
(
α
(
t
)) and
L
(
t
) are defined by
M_{ij}
= (
v_{i}
(
x
),
v_{j}
(x)),
N_{ij}
(
α
) =
A^{β}
(
a
,
u_{h}
:
v_{i}
(
x
),
v_{j}
(
x
)) and
F_{j}
(
α
(
t
)) = (
f
(
x, t, u_{h}
),
v_{j}
(
x
)),
L_{j}
(
t
) =
l_{β}
v_{j}
(
x
)).
For
y
= (
y
_{1}
,
y
_{2}
, · · · ,
y_{m}
)
^{T}
∈ ℝ
^{m}
we let
then
Therefore
M
is a positive definite matrix. By applying the theory on the existence of the solution of the system of the ordinary differential equations, we acquire the existence of the solution of the system (4.3).
Since
f
(
x, t, u
) and
a
(
x, u
) are locally Lipschitz continuous in
u
, −
N
(
α
(
t
))
α
(
t
)+
F
(
α
(
t
)) +
L
(
α
(
t
)) is also locally Lipschitz in
α
(
t
). Thus from the theory on the uniqueness property of the system of the ordinary differential equations the unique existence can be quaranteed locally at (0,
α
(0))
^{T}
.
By the similar analysis we may prove that the uniqueness property of
u_{h}
(
x, t
) holds globally if
a
(
x, t
) and
f
(
x, t, u
) are globally Lipschitz in
t
and
u
.
Remark 1.
From theorem 4.1 we obviously deduce that ∥
u^{h}
(
t
)∥
_{L∞}
is continuous with respect to
t
and
holds for some positive constant
K
^{*}
and sufficiently small
h
provided that
μ
= min(
r
+ 1,
s
) ≥ 1. We define
K
^{*}
satisfying (4.11) which appears in the end this paper as well as (4.4).
Now we let
then
u
−
u_{h}
=
η
+
χ
.
Theorem 4.2.
We assume that the hypotheses of Theorem 3.1 hold. Suppose that f
(
x, t, u
)
and a
(
x, u
)
satisfy that
hold. If
then there is a generic positive constant C such that
where μ
= min(
r
+ 1,
s
).
Proof
. To get the error bound of
u
−
u_{h}
we temporarily assume that
holds for sufficiently small
h
^{*}
. We subtract (4.1) from (4.2) and obtain the following error equation:
from which we have
By (3.1) we have
By (4.5), (4.6), Lemma 3.1 and Theorem 3.1 we have for
ϵ
> 0,
By Theorem 3.1 we get the following estimations:
Therefore
where
C
depends on
and ∥
u
∥
_{s}
. By the assumption (4.5) and (4.6) we get
Now we substitute the estimations of
L
_{1}
∼
L
_{3}
and (4.8) into (4.7) with
v
=
χ
we get for some
Now we choose sufficiently small
ϵ
> 0 and apply the Gronwall inequality to get
Therefore
Now we will verify that we may without loss of generality assume that (4.6) holds.
By (4.4), obviously (4.6) holds for
t
= 0. Suppose that there exist
t
^{*}
such that ∥
u
(
t
) −
u_{h}
(
t
)∥
_{L∞}
< 2
K
^{*}
, ∀
t
<
t
^{*}
but
Now we choose a sequence of
converging to
t
^{*}
. By following the preceding process below (4.6) we obtain the result ∥
χ
(
t_{n}
)∥
^{2}
≤
Ch
^{2μ}
. By applying Lemma 3.1, (3.4) and (4.9) we get
provided that
By Theorem 4.1 we notice that
u_{h}
(
t
) is continuous with respect to
t
, therefore this implies that ∥(
u
−
u_{h}
)(
t
)∥
_{∞}
is continuous with respect to
t
. Hence we get
which contradicts to (4.10). Therefore we may assume that (4.6) holds for any
h
<
h
^{*}
with sufficiently small
h
^{*}
.
BIO
Mi Ray Ohm received her BS degree from Busan National University and Ph.D degree from Busan National University under the direction of Professor Ki Sik Ha. She is a professor at Dongseo University. Her research is centered numerical analysis methods on partial differential equations.
Division of Information Systems Engineering, Dongseo University, 617716, Korea
email : mrohm@dongseo.ac.kr
Hyun Yong Lee received her BS degree from Busan National University and Ph.D degree from University of Tennessee under the direction of Professor Ohannes Karakashian. She is a professor at Kyungsung University. Her research is centered numerical analysis methods on partial differential equations.
Department of Mathematics, Kyungsung University, 608736, Korea
email : hylee@ks.ac.kr
Jun Yong Shin received his BS degree from Busan National University and Ph.D degree from University of Texas under the direction of Professor R. Kannau. He is a professor at Pukyong National University. His research is centered numerical analysis methods on partial differential equations.
Department of Applied Mathematics, Pukyong National University, 608737, Korea
email : jyshin@pknu.ac.kr
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