Fully Discrete Orthogonal Collocation Method of Sobolev Equations

In this paper, the fully discrete orthogonal collocation method for Sobolev equations is considered, and the equivalence for discrete Garlerkin method is proved. Optimal order error estimate is obtained.


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 H 1 -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: In the equations, t u is the time derivative of u, and u ∇ is the gradient of u.

( )
, , a a x y t = and ( )

Fully-Discrete Collocation Method
First, time is divided into n equal parts.Let T t n ∆ = be the time step.Then we introduce the following notations: , , , , .
Then we discrete the spatial region Ω into grids by points ( ) x y are satisfied , is a Bi-cubic Hermit polynomial , , 0 .
Next, we are going to prove existence and uniqueness of collocation solution and obtain the error estimate.

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).
. So ( 2) and ( 4) can be written in the form as fol- , , , , , , , , , where , G D are both matrixs of 4 4 MN MN × and , R S are both vectors of 4MN .
Obviously the solution of equation 0 Fτ = must be satisfied the equation 0 Cτ = , when τ is a vectors of 4MN .So F is nonsingular when C is nonsingular.Then the solutions of ( 2) and ( 4) are unique.To get the existence and uniqueness, we just need to prove singular when t ∆ is sufficiently small.And the nonsingularity of A has been proved [8] in.Thus the theorem is proved.
Next we will need to analyse the error estimate of (4).

Error Estimate
Define interpolation operators ( ) , P P which satisfied the following conditions . 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), ( ) , U x y is the solution of the orthogonal collocation method, and ( ) Ω , then there is the error estimate as follows , .5), the equations Then through the Cauchy inequality, ε-inequality and and the functions a and b are bounded, it leads to the inequality .
The coefficients , K C both have nothing to do with , h t ∆ in the upper eq- uation and following proof.Add the inequality ( 6) and make summation to the series sum from 1 n = to n and multiply t ∆ .Then ( ) is obtained.So it follows from discrete Gronwall lemma that ( ) Then through Cauchy inequality and ε-inequality, ( 6) and ( 7) it leads to the inequality ( ) The results 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 3 , is valid.So it follows from ( 7), ( 9) and (10) that ( ) Thus the theorem is proved.
N.Ma  et al.DOI: 10.4236/jamp.2017.5121922356 Journal of Applied Mathematics and Physics got.It is easy to get K are constants which have nothing to do with h and t η .