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  , the H1-Galerkin Finite Element Method  , the mixed finite element  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  , stochastic PDEs  and reaction diffusion equation  . 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:
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 
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
Now we set the following notations  :
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
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  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
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   , , then there is the error estimate as follows
Proof: First, it is clearly for that
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
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
if is sufficiently small.
At last, let in the second equation of (5), it can be expressed as (7) and (8) implies that
can be obtained from lemma 1.6 in  , where u is sufficiently smooth (C is a positive constant). Moreover (3) in  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.
Sincere thanks to the Basic Subjects Fund and Science Foundations of China University of Petroleum (Beijing) (NO. 2462015YQ0604, NO. 2462015QZDX02).
 Beck, J., Tempone, P. and Tamellini, F.N.L. (2012) On the Optiaml Polynomial Approximation of Stochastic PDEs by Galerkin and Collocation Methods. Mathematical Models & Methods in Applied Sciences, 22, 199-218.
 Parand, K. and Nikarya, M. (2017) A Numerical Method to Solve the 1D and the 2D Reaction Diffusion Equation Based on Bessel Functions and Jacobian Free Newton-Krylov Sub-Space Methods. European Physical Journal-Plus, 132, 496.
 Fernandes, R.I. and Fairweather, G. (1993) Analysis of Alternating Direction Collocation Methods for Parabolic and Hyperbolic Problems in Two Space Variables. Numerical Methods Partial Differential Equations, 9, 191-211.