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-Error Estimates of the Extrapolated Crank-Nicolson Discontinuous Galerkin Approximations for Nonlinear Sobolev Equations
Journal of Inequalities and Applications volume 2010, Article number: 895187 (2010)
Abstract
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 
error estimates of discontinuous Galerkin approximations in both spatial direction and temporal direction.
1. Introduction
Let 
be an open bounded domain in 
, 
with smooth boundary 
, and let 
be given. In this paper, we consider the problem of approximating 
satisfying the following nonlinear Sobolev equations:
where 
denotes the unit outward normal vector to 
and 
is a given function defined on 
. The initial data 
, 
, 
and 
are assumed to be such that (1.1) admits a solution sufficiently smooth to guarantee the convergence results to be presented below. For details 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 
normed space and superconvergence results. Recently, Lin [10] constructed the Galerkin approximation of (1.1) with 
using Crank-Nicolson method and proved the optimal convergence of error in 
normed space. In [11] the authors constructed the semidiscrete finite element approximations of (1.1) with nonlinear boundary condition and obtained the optimal 
-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 
and suboptimally in 
for the energy norm and suboptimally for the 
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 
which are optimal in 
and nearly optimal in 
and they developed a parabolic lift technique for SIPG which leads to 
-optimal and nearly 
-optimal error estimates in 
and negative norms. Recently in [14, 15] Sun and Yang adapted discontinuous Galerkin methods to nonlinear Sobolev equations and obtained the optimal 
error estimates. The main object of this paper is to obtain the optimal 
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 
be a regular quasi-uniform subdivision of 
where 
is a triangle or a quadrilateral if 
and 
is a 
-simplex or 
-rectangle if 
. Let 
and 
. Here, the regular requirement is that there exists a constant 
such that each 
contains a ball of radius 
. The quasi-uniformity requirement is that there is a constant 
such that
We denote the edges (resp., faces for 
) of 
by 
where 
has positive 
dimensional Lebesgue measure, 
, 
, and 
, 
. With each edge (or face) 
, we associate a unit normal vector 
to 
if 
and 
. For 
, 
is taken to be the unit outward vector normal to 
.
For an 
and a domain 
, we denote by 
the Sobolev space of order 
equipped with the usual Sobolev norm 
. We simply write 
instead of 
if 
and 
instead of 
if 
. And also the usual seminorm defined on 
is denoted by 
.
Now for an 
and a given subdivision 
, we define the following space:
For 
with 
, we define the average function 
and the jump function 
such that
where 
with 
.
We associated the following broken norms with the space 
:
where
is an interior penalty term and 
is a discrete positive function that takes the constant value 
on the edge 
and is bounded below by 
and above by 
.
3. Finite Element Spaces and Convergence of Auxiliary Projection
For a positive integer 
, we construct the following finite element spaces:
where 
denotes the set of polynomials of degree less than or equal to 
on 
.
Now we state the following 
-approximation properties and trace inequalities whose proofs can be found in [16, 17].
Lemma 3.1.
Let 
and 
. Then there exist a positive constant 
depending on 
, 
, and 
but independent of 
, 
and 
and a sequence 
, 
, 
, 
such that, for any 
,
where 
and 
is an edge or a face of 
.
Lemma 3.2.
For each 
, there exists a positive constant 
depending only on 
and 
such that the following trace inequalities hold:
where 
is an edge or a face of 
and 
is the unit outward normal vector to 
.
Now we introduce the following bilinear mappings 
and 
defined on 
as
Using the bilinear mappings 
and 
, we construct the weak formulation of problem (1.1) as follows:
Now for a 
we define the following bilinear forms 
and 
on 
such that
and 
satisfy the following boundedness and coercivity properties, respectively. The proofs can be found in [18, 19].
Lemma 3.3.
For a 
, there exists a constant 
satisfying
Lemma 3.4.
For a 
, there exists a constant 
satisfying
Wheeler [20] introduced an elliptic projection to prove the optimal 
-error estimates for Galerkin approximation to parabolic differential equations. Adopting this idea we construct a projection 
such that
By Lemmas 3.3 and 3.4, 
is well defined.
4. The Optimal 
Error Estimates of Fully Discrete Approximations
Error Estimates of Fully Discrete ApproximationsIn this section we construct fully discrete discontinuous Galerkin approximations using extrapolated Crank-Nicolson method and prove the optimal convergence in 
normed space.
For a positive integer 
we let 
and for 
and we define 
and 
. For 
, we define 
, 
and 
.
The extrapolated Crank-Nicolson discontinuous Galerkin approximation 
is defined by
where 
, 
.
To apply (4.1), we need two initial stages 
and 
to be defined in the following:
where 
.
To prove the optimal convergence of 
in 
normed space we denote 
and 
, 
, 
, 
, 
.
Now we state the following approximations for 
whose proofs can be found in [18, 19].
Theorem 4.1.
If 
and 
then there exists a constant 
independent of 
and 
satisfying
(i)
(ii)
.
Theorem 4.2.
If 
, 
, 
and 
then there exists a constant 
independent of 
and 
satisfying
(i)
(ii)
provided that 
.
By simple computations and the applications of Theorem 4.2 we obtain the following lemmas.
Lemma 4.3.
If 
satisfies
then there exists a constant 
independent of 
and 
such that
Consequently from Lemma 4.3 there exists a constant 
independent of
and 
such that
if 
is sufficiently smooth.
Lemma 4.4.
If 
then there exists a constant 
independent of 
and 
such that
Consequently from Lemma 4.4 there exists a constant 
independent of
and 
such that
if 
is sufficiently smooth.
Lemma 4.5.
If we let 
then there exists a constant 
independent of 
and 
such that
Consequently from Lemma 4.5, we induce that there exists a constant 
independent of
and 
such that
if 
is sufficiently smooth.
Theorem 4.6.
For 
and 
, if 
, 
then there exists a constant 
independent of
and 
such that for 
, 
, 
, 
hold where 
and 
.
Proof.
From (4.1) and (1.1), we have
By the notations of 
and 
, we get
By the definition of 
, we obtain
From the definition of 
, we have
Substituting (4.12)–(4.14) in (4.11) and choosing 
imply that
By Cauchy-Schwarz'sinequality clearly we have
By the definition of 
we have
For the definition of 
we get
Applying (4.17) and (4.18) in (4.15) we conclude that
For sufficiently small 
by applying Lemma 4.3 there exists a constant 
such that
Applying Lemmas 4.3 and 4.4, 
can be estimated as follows:
We obtain the following estimates of 
for each 
From the definition of 
, we can separate 
as follows:
By applying Lemma 4.5, 
can be estimated in the following way:
Similarly there exists a constant 
such that
By applying the trace inequality we have
From the estimation of 
, 
, we have
By applying Lemma 4.3 we obviously obtain
Now we can separate 
as follows:
Since
can be estimated as follows
We apply Lemma 3.2 to estimate 
as follows:
From the result of approximation of 
of Theorem 4.1
Therefore we get
Similarly, 
and 
are estimated as follows:
Substituting the estimates of 
, 
into (4.19), we get
If we sum both sides of (4.36) from 
to 
, then we obtain
which implies
where 
is sufficiently small. By applying the discrete version of Gronwall's inequality, we have
Therefore by applying the result of Lemma 4.7 we have
which proves the optimal 
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 
and 
, if 
, 
and 
for some constant 
then there exists a constant 
independent of
and 
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Acknowledgment
This research was supported by Dongseo University Research Grants in 2009.
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Ohm, M., Lee, H. & Shin, J. 
-Error Estimates of the Extrapolated Crank-Nicolson Discontinuous Galerkin Approximations for Nonlinear Sobolev Equations.
J Inequal Appl 2010, 895187 (2010). https://doi.org/10.1155/2010/895187
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Keywords
- Galerkin Method
- Constant Independent
- Spatial Discretization
- Discrete Method
- Discrete Version
