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 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 -norm error estimate is derived. Just because of a damping term 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.
The time discontinuous Galerkin method was proposed by Delfour in 1981  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 , 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 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 , Lesaint and Raviart  and Jamet  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. [5–9]).
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 , the problem of moisture migration in soil , 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  and the nonlinear Sobolev equations in [17–20]. The differences-streamline diffusion method and DG finite element method were also discussed for Sobolev equations in [21–23], 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 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 -norm error estimate is derived. Finally, we describe conclusions and perspectives in Section 6.
2. Mathematical Model
We consider the following quasi-linear Sobolev equations: to find such thatwhere is a convex domain in () with a smooth boundary .
We consider the coefficients in (1) with the following conditions (I):(i), .(ii)Assume , , , , . There exist positive constants , , , and satisfying(iii)Assume that and satisfy the uniform Lipschitz condition with respect to the variable . Let have uniform bounded derivatives of order up to 1 and 2 with respect to the variable and variable , respectively.
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 , .
3. 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 such that
Let be an unnecessarily uniform subdivision of , and
Let be a partition of , let be the diameter of element , and . Define as a finite element space with index (cf. ). We want to find an approximate solution of (1) belonging to the spaceIn other words, a function in reduces to a polynomial of degree at most in on each interval with coefficients in . Note that these functions in are allowed to be discontinuous at the nodes , , and are taken to be continuous to the left side of the nodes. For , we denote and as the value of and its limit from above , respectively. We write for the restrictions to of the function in .
Our time discontinuous Galerkin finite element scheme is to find such thatwith an initial valueHere, is a certain approximation to in .
Noticing that there existwe can rewrite (8) as
Hence, on each time interval , we have a local scheme: to find , , 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 , or scalars if the dimension of is , and the number of equations, which is equal to the dimension of , is also . Therefore, the main advantage of our scheme is that the approximate solution can be solved in each time interval gradually.
4. 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 bywhere , .
Obviously, for any , the above equation has a unique solution so that the projection is uniquely identified in . 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 , we define a setIt is easy to see that is a closed set in .
Step 1 (we prove that when is small enough, maps onto itself). Taking in (13), we can obtainwhere the norm , .
Applying the -inequality, we findwhere is the Lipschitz constant of with respect to .
Combining (15) with (16) shows(2) Taking in (13), since , we haveNotice that there existsHence, inserting (19) into (18) yieldsUsing the -inequality to (20), we obtainTherefore, by taking in (21), the following estimate holds:(3) It is easy to see that there existsAs and are both polynomials in time of degree in , and the norms are equivalent on finite dimensional space, we know that there exists a constant , such thatThen, it follows from (23) and (24) thatFurthermore, (25) can be amplified asPutting (17) and (22) into the right-hand side of (26) and usingwe haveIf is taken to be small enough to satisfy (i) and (ii) the last term on the right-hand side of (28) , we derivethat is, .
Step 2 (we prove that when is small enough, is continuous on ). For any point , , denote , . Then, from (13) we haveTaking in (30) and with similar analyses in Step 1, we can find that when is small enough, there existsThis 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.
5. -Norm Error Estimate
In order to derive an optimal -norm error estimate, it is necessary to refer to a nonstandard elliptic projection from  such thatwhere constant . Also, we have the following lemmas about projection .
Lemma 3 (see ). There exists a unique solution for any .
Lemma 4 (see ). Assume that and , , , , , and are bounded. Then, there exists a positive constant such that
Lemma 5 (see ). Assume that , and , , , and are bounded. There exists a positive constant such that
Lemma 7. Let , where satisfies (36). Then, there existwhere , , and ; denotes the th-order derivative with respect to ; is a positive constant depending only on .
Defining , , then we have . Taking them into (38), we derive
From the two inequalitieswe know that
So, it remains to estimate and . From now on, we use notations and .
Lemma 8. Let , , satisfying , , . Define to satisfyThen, we can have the following estimates:
Proof. () Taking in (44), since , we can getFurther, we haveBecause and are both polynomials in of degree at most , we have two inverse inequalitiesCombining the above inequalities with (49), we obtain This is just estimate (45).
() Now, we go to prove estimate (46). Taking into (44), we have From (53), we conclude thatCombining (54) with (45), taking to be small enough, and using the Gronwall lemma, we can derive estimate (46).
() Finally, we prove estimate (47). Define a bilinear form: ,From (44), we know that , , and defined in Lemma 8 satisfy Let . We introduce an auxiliary function , which satisfies andUsing (55), we know where satisfies the conditions in Lemma 8.
Hence, from (45) and (46), we can derive the estimateNow, we consider an adjoint problem of (1): to find such thatIn other words, that is the finite element solution in of Similar to the derivation of result (59), we can bound the estimateTaking in (60), we haveWe analyze the terms on the right-hand side of (63) term by term. Using (46) and (62), we haveUsing (56) and (62), we haveSubstituting (64) and (65) into (63) yieldsBy the arbitrariness of , taking into (66), we get estimate (47).
In order to obtain the optimal order estimate with respect to , we still need the following lemma (Lemma 9).
Lemma 9. Under the conditions in Lemma 8, one has