The Time Discontinuous <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06570053pt" id="M1" height="16.0088pt" version="1.1" viewBox="-0.0657574 -15.9431 21.9869 16.0088" width="21.9869pt"><g transform="matrix(.018,0,0,-0.018,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.013,0,0,-0.013,15.027,-7.899)"><path id="g50-50" d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"/></g></svg>-Galerkin Mixed Finite Element Method for Linear Sobolev Equations

Yu, Hong; Sun, Tongjun; Li, Na

doi:https://doi.org/10.1155/2015/618258

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2015 | Article ID 618258 | https://doi.org/10.1155/2015/618258

The Time Discontinuous -Galerkin Mixed Finite Element Method for Linear Sobolev Equations

Hong Yu,¹Tongjun Sun,²and Na Li¹

Academic Editor: Alicia Cordero

Received14 Dec 2014

Revised18 Mar 2015

Accepted19 Mar 2015

Published02 Apr 2015

Abstract

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.

1. Introduction

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 [1], the transport problems of humidity in soil [2], 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 [12] 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 [13] and Lesaint and Raviart [14] 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 [20], 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 [21] 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 [22], the second-order hyperbolic equations [23], and Sobolev equations [8]. Recently, some researchers applied this method to other types of problems, such as RLW equation [24], heat transport equations [25], and pseudo-parabolic equation [26].

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

Then, the variational form of (3a) and (3b) is to find such thatwhere , .

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 , .

Let

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.

Hence, the time discontinuous -Galerkin mixed finite element scheme of (6a) and (6b) is to find such thatHere, we take and is a certain initial approximation to in .

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

In this section, we prove the existence and uniqueness of the solutions of (18)-(19). First, we adopt the following lemma [27].

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 .

Proof. Let , . Using (22), we haveMultiplying both sides of (24) by leads toIntegrating (25) in , , and using , we can obtainSubstituting (26) into (22) completes the existence of (23).

By Lemma 2, we can have the following theorem.

Theorem 3. The solutions of (18)-(19) are existent and unique.

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 [28]. These proofs are different from well-known Brouwer’s fixed point theorem [29], 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

Now, we turn to consider the error estimates of (18)-(19). In order to derive -norm error estimates, we adopt the nonstandard elliptic projection such that [9]where constant .

Then, we have some properties of as the following lemmas.

Lemma 4 (see [9]). There exists a unique solution of (37), for any .

Lemma 5 (see [9]). Let . Assume that , , and , , , and are bounded. Then, there exists a positive constant such that

Lemma 6 (see [9]). Assume that , , , and are bounded. Then, there exists a positive constant such that

Lemma 7 (see [9]). If the order of approximate accuracy of space is , there exists a positive constant such that

Let be the standard finite element interpolation of and . By the Lagrange interpolation theory [28, 30], we have the following.

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:

By the Lagrange interpolation theory [28, 30], the following holds.

Lemma 9. Let and , where satisfies (42) and satisfies (43). Then, there existwhere , , , and denotes the th order derivative with respect to and is a positive constant depending only on .

Let and . Then, we have the decompositions

Combining (6a) and (6b) with (18)-(19), we achieve the error equations

In order to get the optimal order estimation of , we still need some lemmas. First of all, it is Lemma from [31]. 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.

Lemma 12. Using Lemmas 10 and 11, we can get the following estimates for the error equations (46)-(47):

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.

Finally, by the triangle inequality and Lemmas 5, 7, 8, 9, and 12, we can establish the following theorem.

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.

5. Conclusions

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.

Acknowledgments

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.

References

G. I. Barenblatt, I. P. Zheltov, and I. N. Kochina, “Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks,” Journal of Applied Mathematics and Mechanics, vol. 24, no. 5, pp. 1286–1303, 1960.
View at: Publisher Site | Google Scholar
D. M. Shi, “The initial-boundary value problem for a nonlinear equation of migration of moisture in soil,” Acta Mathematicae Applicatae Sinica, vol. 13, no. 1, pp. 31–38, 1990.
View at: Google Scholar | MathSciNet
T. W. Ting, “A cooling process according to two-temperature theory of heat conduction,” Journal of Mathematical Analysis and Applications, vol. 45, pp. 23–31, 1974.
View at: Publisher Site | Google Scholar | MathSciNet
P. J. Chen and M. E. Gurtin, “On a theory of heat conduction involving two temperatures,” Zeitschrift für Angewandte Mathematik und Physik ZAMP, vol. 19, no. 4, pp. 614–627, 1968.
View at: Publisher Site | Google Scholar
T. Sun and D. Yang, “The finite difference streamline diffusion methods for Sobolev equations with convection-dominated term,” Applied Mathematics and Computation, vol. 125, no. 2-3, pp. 325–345, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
T. J. Sun and D. P. Yang, “A priori error estimates for interior penalty discontinuous Galerkin method applied to nonlinear Sobolev equations,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 147–159, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
T. J. Sun, “Streamline-diffusion F.E.M. for Sobolev equations with convection-dominated term,” Applied Mathematics. A Journal of Chinese Universities. English Series, vol. 16, no. 1, pp. 63–71, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
L. Guo and H. Z. Chen, “ $H^{1}$ -Galerkin mixed finite element method for the Sobolev equation,” Journal of Systems Science and Mathematical Sciences, vol. 26, no. 3, pp. 301–314, 2006.
View at: Google Scholar
Y. P. Lin and T. Zhang, “Finite element methods for nonlinear Sobolev equations with nonlinear boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 165, no. 1, pp. 180–191, 1992.
View at: Publisher Site | Google Scholar | MathSciNet
J. S. Zhang, D. P. Yang, and J. Zhu, “Two new least-squares mixed finite element procedures for convection-dominated Sobolev equations,” Applied Mathematics, Series B, vol. 26, no. 4, pp. 401–411, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Liu, H. Li, S. He, W. Gao, and S. Mu, “A new mixed scheme based on variation of constants for Sobolev equation with nonlinear convection term,” Applied Mathematics. A Journal of Chinese Universities. Series B, vol. 28, no. 2, pp. 158–172, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
M. Delfour, W. Hager, and F. Trochu, “Discontinuous Galerkin methods for ordinary differential equations,” Mathematics of Computation, vol. 36, no. 154, pp. 455–473, 1981.
View at: Publisher Site | Google Scholar | MathSciNet
P. Jamet, “Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain,” SIAM Journal on Numerical Analysis, vol. 15, no. 5, pp. 912–928, 1978.
View at: Publisher Site | Google Scholar | MathSciNet
P. Lasaint and P.-A. Raviart, “On a finite element method for solving the neutron transport equation,” in Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, Ed., pp. 89–123, Academic Press, New York, NY, USA, 1974.
View at: Google Scholar | MathSciNet
K. Eriksson, C. Johnson, and V. Thomée, “Time discretization of parabolic problems by the discontinuous Galerkin method,” RAIRO Modélisation Mathématique et Analyse Numérique, vol. 19, no. 4, pp. 611–643, 1985.
View at: Google Scholar | MathSciNet
K. Eriksson and C. Johnson, “Adaptive finite element methods for parabolic problems. I. A linear model problem,” SIAM Journal on Numerical Analysis, vol. 28, no. 1, pp. 43–77, 1991.
View at: Publisher Site | Google Scholar | MathSciNet
K. Eriksson and C. Johnson, “Adaptive finite element methods for parabolic problems II. Optimal error estimates in $L_{\infty} L_{2}$ and $L_{\infty} L_{\infty}$ ,” SIAM Journal on Numerical Analysis, vol. 32, no. 3, pp. 706–740, 1995.
View at: Publisher Site | Google Scholar | MathSciNet
K. Eriksson and C. Johnson, “Adaptive finite element methods for parabolic problems. IV. Nonlinear problems,” SIAM Journal on Numerical Analysis, vol. 32, no. 6, pp. 1729–1749, 1995.
View at: Publisher Site | Google Scholar | MathSciNet
Ch. G. Makridakis and I. Babuska, “On the stability of the discontinuous Galerkin method for the heat equation,” SIAM Journal on Numerical Analysis, vol. 34, no. 1, pp. 389–401, 1997.
View at: Publisher Site | Google Scholar | MathSciNet
A. I. Pehlivanov, G. F. Carey, and R. D. Lazarov, “Least-squares mixed finite elements for second-order elliptic problems,” SIAM Journal on Numerical Analysis, vol. 31, no. 5, pp. 1368–1377, 1994.
View at: Publisher Site | Google Scholar | MathSciNet
A. K. Pani, “An $H^{1}$ -Galerkin mixed finite element method for parabolic partial differential equations,” SIAM Journal on Numerical Analysis, vol. 25, pp. 712–727, 1998.
View at: Google Scholar
A. K. Pani and G. Fairweather, “H¹-Galerkin mixed finite element methods for parabolic integro differential equations,” IMA Journal of Numerical Analysis, vol. 22, no. 2, pp. 231–252, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
A. K. Pani, R. K. Sinha, and A. K. Otta, “An $H^{1}$ -Galerkin mixed finite element methods for second order hyperbolic equations,” International Journal of Numerical Analysis and Modeling, vol. 1, pp. 111–129, 2004.
View at: Google Scholar
Y. Liu, H. Li, Y. W. Du, and J. F. Wang, “Explicit multistep mixed finite element method for RLW equation,” Abstract and Applied Analysis, vol. 2013, Article ID 768976, 12 pages, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
Z. J. Zhou, “An $H^{1}$ -Galerkin mixed finite element method for a class of heat transport equations,” Applied Mathematical Modelling, vol. 34, no. 9, pp. 2414–2425, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Liu and H. Li, “H¹-Galerkin mixed finite element methods for pseudo-hyperbolic equations,” Applied Mathematics and Computation, vol. 212, no. 2, pp. 446–457, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
M. Y. Huang, The Numerical Methods of Evolution Equations, Science Press, Beijing, China, 2004, (Chinese).
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing, Amsterdam, The Netherlands, 1978.
View at: MathSciNet
F. E. Browder, “Existence and uniqueness theroems for sloutions of nonlinear boundary value problems,” in Applications of Partial Differential Equations, R. Finn, Ed., pp. 24–29, Amercian Mathematical Society, Providence, RI, USA, 1965.
View at: Google Scholar
M. F. Wheeler, “A priori $L^{2}$ error estimates for Galerkin approximations to parabolic partial differential equations,” SIAM Journal on Numerical Analysis, vol. 10, no. 4, pp. 723–758, 1973.
View at: Publisher Site | Google Scholar | MathSciNet
H. Yu, The time discontinuous Galerkin finite element methods for the Sobolev equations [M.S. thesis], Shandong University, Shandong, China, 2009.

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Copyright © 2015 Hong Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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