The Time Discontinuous -Galerkin Mixed Finite Element Method for Linear Sobolev Equations
We combine the -Galerkin mixed finite element method with the time discontinuous Galerkin method to approximate linear Sobolev equations. The advantages of these two methods are fully utilized. The approximate schemes are established to get the approximate solutions by a piecewise polynomial of degree at most with the time variable. The existence and uniqueness of the solutions are proved, and the optimal -norm error estimates are derived. We get high accuracy for both the space and time variables.
We consider the following linear Sobolev equations to find such thatwhere is a convex domain in with a smooth boundary . Functions , , , and are known. Assume that and have bounded derivatives with respect to the variables and , respectively, and satisfywhere , , , , and are positive constants.
There has been a wide range of practical applications of Sobolev equations in engineering, such as the porous theories concerned with percolation into rocks with cracks , the transport problems of humidity in soil , and the heat conduction problems between different media [3, 4]. So many numerical simulation methods have been proposed to solve these equations; for example, see [5–11].
The time discontinuous Galerkin method was proposed by Delfour et al. in 1981  to analyze the ordinary differential equations. The purpose of this method is to find the approximate solutions as a piecewise polynomial of degree at most with time variable by discretizing the time variable with Galerkin method. So the similar processing method can be used to analyze the space and time variables in the definition and analysis. This method considered the space and time variables together in order to utilize the advantages of the finite element method on both the space and time variables. Hence, the accuracy of numerical solutions is improved. Jamet  and Lesaint and Raviart  used the time discontinuous Galerkin method to study the partial differential equations. Eriksson et al. promoted and enriched this method further; for example, see [15–19].
It is well known that standard mixed finite element spaces have to satisfy the LBB-consistency condition (also called Inf-Sup condition), which restricts the choice of finite element spaces. For example, in , the Raviart-Thomas spaces of index are used for the second-order elliptic problems.
In order to circumvent the LBB-consistency condition, -Galerkin mixed finite element method was introduced by Pani  to parabolic equations. By introducing a flux variable of the primary variable , parabolic equations are changed into a first-order system. Then, -Galerkin finite element method is used to approximate the system. The finite element spaces for variable and flux are allowed to be of different polynomial degrees and not subject to the LBB-consistency condition. Hence, the obtained estimations can distinguish the better approximation properties of and . Moreover, the quasi-uniformity condition was not imposed on the finite element mesh for and -norm error estimates. More applications of -Galerkin mixed finite element method had been done to, for example, the parabolic integro-differential equations , the second-order hyperbolic equations , and Sobolev equations . Recently, some researchers applied this method to other types of problems, such as RLW equation , heat transport equations , and pseudo-parabolic equation .
In this paper, we combine -Galerkin mixed finite element method and the time discontinuous Galerkin method to approximate linear Sobolev equations. We establish the time discontinuous -Galerkin mixed finite element schemes and expect to utilize the advantages of the two above-mentioned methods to obtain a high-accuracy numerical method.
The rest of this paper is organized as follows. In Section 2, we present the time discontinuous -Galerkin mixed finite element schemes. In Section 3, we prove the existence and uniqueness of the solutions. In Section 4, the optimal -norm error estimates are derived. In Section 5, we draw some conclusions of this paper.
2. Time Discontinuous -Galerkin Mixed Finite Element Schemes
In this section, we establish the approximate schemes for (1a), (1b), and (1c). Let us introduce an intermediate variable . Then, (1a), (1b), and (1c) can be rewritten as the following first-order system:
Let Sobolev space , equipped with the norm and inner product defined byrespectively.
Similarly, let , . The norms on these two spaces are defined, respectively, by
Clearly (6a) is obtained by multiplying (3a) by and integrating the resulting equation with respect to . Multiplying (3b) by and integrating the first term by parts givewhere the condition is used. From (3a), we know . Substituting this expression into the above equation yields (6b).
Assume that and are the finite dimensional subspaces of and , respectively, with the following approximation properties: for , positive integers
Let be an unnecessarily uniform subdivision of and
With a given positive integer , we will look for the approximation solutions of (6a) and (6b), which reduce to a polynomial of degree at most with time variable on each subinterval with coefficients in and , respectively. That is to say, they belong to the following finite element spaces:Note that the functions of these two spaces are allowed to be discontinuous at the time nodal points but are taken to be continuous to the left there. For , we denote , .
Notice that Here, we do not consider the continuity of at in the above equation, which means the time discontinuous Galerkin finite element space can be adopted.
Since(14) can be rewritten as
Due to the discontinuities of and with respect to , the local forms of (17) and (15) can be given as follows:The advantage of the local forms (18)-(19) is that they can be solved locally in each time interval gradually.
3. Existence and Uniqueness of the Solutions
Lemma 1 (Gronwall’s inequality). Assume that is a continuous function in satisfyingwhere and . Then, there holds
Similarly, we have the following.
Lemma 2. Assume is a nonnegative real number, , is a continuous function in that satisfies , andThen, there existswhere .
By Lemma 2, we can have the following theorem.
Proof. Since (18)-(19) are linear equations about and , the existence and uniqueness of the solutions are equivalent to the homogenous linear equations only that have zero solutions on each time interval . These proofs are different from well-known Brouwer’s fixed point theorem , which is used to consider the nonlinear boundary problems.
Letting , and taking in (18), we holdUsing Cauchy inequality, Poincaré inequality, and (2), we haveThis yieldsUsing Lemma 2, we deriveTaking in (19), we findUsing Cauchy inequality, -inequality, and (2), we haveTherefore, by taking in (32), we obtainTaking (33) into (30) leads toThen, using Lemma 2, we know . That is to say, . So we can obtainTaking (35) into the following inequalitywe obtain ; that is, . Then, using Poincaré inequality, we have ; that is, .
Finally, taking in (33), we have ; that is to say, . So the existence and uniqueness of the solutions are proved.
4. -Norm Error Estimates
Then, we have some properties of as the following lemmas.
Lemma 5 (see ). Let . Assume that , , and , , , and are bounded. Then, there exists a positive constant such that
Lemma 6 (see ). Assume that , , , and are bounded. Then, there exists a positive constant such that
Lemma 7 (see ). If the order of approximate accuracy of space is , there exists a positive constant such that
Lemma 8. If the order of approximate accuracy of space is , there exists a positive constant such that
Assume , are the following interpolations of and with respect to , respectively:
Let and . Then, we have the decompositions
In order to get the optimal order estimation of , we still need some lemmas. First of all, it is Lemma from . The proof of this lemma is so long that we only adopt it here for the sake of brevity.
Lemma 10. Let , , ; . They satisfy , , andThen, we have the following estimates:
Lemma 11. Under the conditions in Lemma 10, there also exist
Proof. By Lemma 10, we know that there existTaking in (47) and letting be small enough, we can getTaking (53) into (55) and letting be small enough, we obtain (52).
Then, taking (55) back into (53) and (54) and also letting be small enough, we finally achieve (50) and (51), respectively. The proof of Lemma 11 is completed.
Proof. (1) Using (50), we haveThen, using Lemmas 5, 8, and 9, we can deriveTaking the above estimates into (59) leads toThen, using the Poincaré inequality , , we obtain the estimate (56).
(2) For , we knowFurther, use (51) and (59) to getSimilar to the analysis of getting (56), we obtain the estimate (57).
(3) Similarly, using (52), (56) and Lemmas 8 and 9, we haveSo, we obtain the estimate (58). Then, the proof of Lemma 12 is completed.
Theorem 13. Let be the solution of the original problem (1a), (1b), and (1c); and are the finite element solutions of the approximate schemes (14)-(15). Assume that is smooth enough to satisfy the required regularities in the analyses, and . Then, we have the following error estimates:
From this theorem, we see that with respect to space variable the optimal -norm error estimates of both and are derived. Also, -order accuracy for time variable is obtained. That is to say, we get high accuracy for both the space and time variables.
In this paper, we have solved the linear Sobolev equations (1a), (1b), and (1c) by the time discontinuous -Galerkin mixed finite element method. We have presented the approximate schemes, proved the existence and uniqueness of the solutions, and derived the optimal -norm error estimates for both the primary variable and intermediate variable . In particular, high-order accuracy for time variable also is obtained. In our forthcoming work, some numerical experiments will be considered to verify our analysis for the schemes.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The second author is supported by the NSF of China (nos. 11271231 and 11301300). The authors would like to thank the anonymous referees for their comments and suggestions on the paper.
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