A Time Discontinuous Galerkin Finite Element Method for Quasi-Linear Sobolev Equations

We present a time discontinuous Galerkin finite element scheme for quasi-linear Sobolev equations. The approximate solution is sought as a piecewise polynomial of degree in time variable at most q − 1 with coefficients in finite element space. This piecewise polynomial is not necessarily continuous at the nodes of the partition for the time interval. The existence and uniqueness of the approximate solution are proved by use of Brouwer’s fixed point theorem. An optimal L∞(0, T;H(Ω))-norm error estimate is derived. Just because of a damping term u xxt included in quasi-linear Sobolev equations, which is the distinct character different from parabolic equation, more attentions are paid to this term in the study. This is the significance of this paper.


Introduction
The time discontinuous Galerkin method was proposed by Delfour in 1981 [1] to analyze the ordinary differential equations.As well known, the fully discrete scheme for unsteady partial differential equations is normally derived by first discretizing in the space variables by the Galerkin finite element method and then replacing the resulting system of ordinary differential equation with respect to time by the finite difference method in the time variable.In [2], the time discontinuous Galerkin method was presented to approximate the heat conduction equation.By applying the Galerkin method to time variable, the purpose of this method was to find the approximate solution as a piecewise polynomial of degree in time variable  at most  − 1 with coefficients in finite element space.The piecewise polynomial was not necessarily continuous at the nodes of the partition for the time interval.So, a similar processing method can be used to analyze the space and time variables in the definition and analysis.Unifying the space and time variables, time discontinuous finite element method overcomes the low order accuracy in traditional finite element method caused by the difference discretization in time.This method has high order accuracy in space and time directions, good dissipation on unstructured mesh, and unconditional stability.Thus, it becomes an efficiency method for the problems dependent on time.It has been successfully used in the fields of fluid mechanics, heat conduction, elastic dynamics, and structural mechanics.
Prior to the work of [2], Lesaint and Raviart [3] and Jamet [4] used such time discontinuous Galerkin finite element method as mentioned above to study the partial differential equations, respectively.After that, many works were analyzed about both the theoretical results and use of the time discontinuous Galerkin finite element method, such as the works of Eriksson, Johnson, Thomée, and Babuska (cf.[5][6][7][8][9]).
Sobolev equations are one of the important partial differential equations in practical use.They arise in the percolation theory of fluid through the crack of rock [10], the problem of moisture migration in soil [11], the problem of heat conduction between different media [12,13], and so on.The existence and uniqueness of the solution of the Sobolev equation were discussed in [14,15].Numerical treatments for Sobolev equations can be found in many papers.The finite element method was discussed for the semilinear Sobolev equations in [16] and the nonlinear Sobolev equations in [17][18][19][20].The differences-streamline diffusion method and DG finite element method were also discussed for Sobolev equations in [21][22][23], respectively.
In this paper, we will establish a time discontinuous Galerkin finite element scheme for the quasi-linear Sobolev equations.That is to say, the approximate solution will be sought as a piecewise polynomial in time variable of degree at most  − 1 with coefficients in finite element space, which is not necessarily continuous at the nodes of the partition for the time interval.More attentions will be paid to treating a damping term   , which is a distinct character of Sobolev equations different from parabolic equation.To our knowledge, this paper appears to be the first trial to approximate quasi-linear Sobolev equations by using the time discontinuous Galerkin finite element method.
The rest of this paper is organized as follows.In Section 2, we briefly introduce the quasi-linear Sobolev equations and some assumptions.In Section 3, we present the time discontinuous Galerkin finite element scheme.In Section 4, we prove the existence and uniqueness of the approximate solution by use of Brouwer's fixed point theorem.In Section 5, an optimal  ∞ (0, ;  1 (Ω))-norm error estimate is derived.Finally, we describe conclusions and perspectives in Section 6.
For the purpose of theoretical analyses, we need the following regularity assumptions on the solution of ( 1): (II) where   = / and    denotes the th-order derivative with respect to ,  ≥ 1.

Time Discontinuous Galerkin Finite Element Scheme
In this section, we establish the approximate scheme based on the time discontinuous Galerkin method for (1).First, we introduce two bilinear forms With the above notations, the variational form of (1) can be rewritten as follows: to find () ∈  1 0 (Ω) such that Let 0 =  0 <  1 < ⋅ ⋅ ⋅ <   =  be an unnecessarily uniform subdivision of [0, ], and Let T ℎ = {} be a partition of Ω, let ℎ  be the diameter of element , and ℎ = max ∈T ℎ ℎ  .Define  ℎ,0 ⊂  1 0 (Ω) as a finite element space with index  (cf.[24]).We want to find an approximate solution of (1) belonging to the space In other words, a function in  ℎ, reduces to a polynomial of degree at most  − 1 in  on each interval   with coefficients in  ℎ,0 .Note that these functions in  ℎ, are allowed to be discontinuous at the nodes   ,  = 0, . . .,  − 1, and are taken to be continuous to the left side of the nodes.For  ∈  ℎ, , we denote   and   + as the value of  and its limit from above   , respectively.We write   ℎ, for the restrictions to   of the function in  ℎ, .
Our time discontinuous Galerkin finite element scheme is to find  ∈  ℎ, such that with an initial value Here,  0,ℎ is a certain approximation to  0 in  ℎ,0 .
Noticing that there exist we can rewrite (8) as Hence, on each time interval   , we have a local scheme: to find  ∈   ℎ, ,  = 0, 1, . . .,  − 1, such that Remark 1. From scheme (12), we can notice that the solution on   can be determined once   and   + are given.In fact, the solution is defined on   by  elements in  ℎ,0 , or  ℎ scalars if the dimension of  ℎ,0 is  ℎ , and the number of equations, which is equal to the dimension of   ℎ, , is also  ℎ .Therefore, the main advantage of our scheme is that the approximate solution can be solved in each time interval gradually.

Existence and Uniqueness of the Solution
In this section, we prove the existence and uniqueness of solution (12).We can obtain the following theorem.
Theorem 2. When   is small enough, the solution of ( 12) exists uniquely.
Proof.Let us introduce a projection  :   ℎ,  →   ℎ, by where  = ,  ∈   ℎ, .Obviously, for any  ∈   ℎ, , the above equation has a unique solution so that the projection  is uniquely identified in   ℎ, .If  has a fixed point,  * =  * , then  =  * is the solution of (12).If  has a unique fixed point, the solution of ( 12) is unique.Now, we begin to prove that  has a unique fixed point by three steps.For a real number  > 0, we define a set It is easy to see that   is a closed set in   ℎ, .
Step 2 (we prove that when   is small enough,  is continuous on   ).For any point  0 ∈   ,  ∈   , denote  0 =  0 ,  = .Then, from (13) we have Taking V =  −  0 in (30) and with similar analyses in Step 1, we can find that when   is small enough, there exists lim This means  is continuous on   .
Step 3. The above proofs show that  is a contraction mapping on   .Using Brouwer's fixed point theorem, we know that  has a unique fixed point in   ; that is,  * =  * .Therefore, the existence and uniqueness of the solution  =  * of ( 12) have been proved.Thus, the proof of Theorem 2 is completed.
In order to obtain the optimal order estimate with respect to , we still need the following lemma (Lemma 9).Lemma 8,

Conclusions
The quasi-linear Sobolev equations (1) are solved by the time discontinuous Galerkin finite element methods.We presented the approximate scheme and proved the existence and uniqueness of the approximate solution by use of Brouwer's fixed point theorem.We made the error analysis and derived the optimal  ∞ (0, ;  1 (Ω))-norm error estimates.We got higher accuracy for both the space variable and the time variable in this paper.In the forthcoming work, some numerical experiments will be considered to verify our analysis for the schemes.