Fully Discrete Orthogonal Collocation Method of Sobolev Equations ()
1. Introduction
Sobolev equations are a class of mathematical physics equations, which are widely used in engineering field. Many numerical methods have been proposed, such as the characteristic difference method [1] , the H1-Galerkin Finite Element Method [2] , the mixed finite element [3] and so on. The collocation method now is widely used in many fields including engineering technology and computational mathematics. Many applications have been proved effectively, e.g. the heat conduction equation [4] , stochastic PDEs [5] and reaction diffusion equation [6] . The collocation method has high convergence order and does not need to calculate numerical integration so that the calculation is simple. So now we consider the application of fully discrete collocation method for Sobolev equations. We consider the linear Sobolev equations as follows:
(1)
In the equations,
is the time derivative of u, and
is the gradient of u.
,
is the border of
.
and
are known bounded differentiable functions.
2. Fully-Discrete Collocation Method
First, time is divided into n equal parts. Let
be the time step. Then we
introduce the following notations:
Then we discrete the spatial region
into grids by points
and
are satisfied
. Let [7]
The four Gauss points
in
are collocation points as follows:
, where
,
. Then the intermediate variable
is introduced so that the orthogonal collocation scheme as follows can be established. Seeking
, such that
(2)
Now we set the following notations [4] :
(3)
Next, we are going to prove existence and uniqueness of collocation solution and obtain the error estimate.
3. Discrete Galerkin Method
Consider the following discrete Galerkin scheme
(4)
Theorem 3.1: The solutions of (4) and (2) are equivalent, existent and unique.
Proof: From the Equation (3), it is clear that the solution of (2) must be the solution of (4).
Let
,
be a group base of
. Thereupon
can be expressed as
. So (2) and (4) can be written in the form as follows
where
are both matrixs of
and
are both vectors of
. Obviously the solution of equation
must be satisfied the equation
, when
is a vectors of
. So
is nonsingular when
is nonsingular. Then the solutions of (2) and (4) are unique. To get the existence and uniqueness, we just need to prove
where
is nonsingular when
is sufficiently small. And the nonsingularity of
has been proved [8] in. Thus the theorem is proved.
Next we will need to analyse the error estimate of (4).
4. Error Estimate
Define interpolation operators
which satisfied the following conditions
i.e.,
. Now we can get the error equations
(5)
where
. Then there is the theorem as follows.
Theorem 4.1: If u(x,y) is the accurate solution of (1),
is the solution of the orthogonal collocation method, and
satisfies the condition [4] [7]
,
, then there is the error estimate as follows
Proof: First, it is clearly for
that
(6)
Then let
in (5), the equations
can be got. It is easily calculated to see that
Then through the Cauchy inequality, ε-inequality and
,
and the functions a and b are bounded, it leads to the inequality
The coefficients
both have nothing to do with
in the upper equation and following proof. Add the inequality (6) and make summation to the series sum from
to
and multiply
. Then
is obtained. So it follows from discrete Gronwall lemma that
(7)
if
is small enough.
Second, let
in (5), the equations
can be got. It is easy to get
Then through Cauchy inequality and ε-inequality, (6) and (7) it leads to the inequality
(8)
if
is sufficiently small.
At last, let
in the second equation of (5), it can be expressed as
(7) and (8) implies that
(9)
The results
(10)
can be obtained from lemma 1.6 in [4] , where u is sufficiently smooth (C is a positive constant). Moreover (3) in [7] implies that
is valid. So it follows from (7), (9) and (10) that
where
and
are constants which have nothing to do with
and
. Thus the theorem is proved.
Acknowledgements
Sincere thanks to the Basic Subjects Fund and Science Foundations of China University of Petroleum (Beijing) (NO. 2462015YQ0604, NO. 2462015QZDX02).