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The Scientific World Journal
Volume 2013 (2013), Article ID 756281, 11 pages
http://dx.doi.org/10.1155/2013/756281
Research Article

A New Linearized Crank-Nicolson Mixed Element Scheme for the Extended Fisher-Kolmogorov Equation

1School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China
2School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

Received 26 April 2013; Accepted 3 June 2013

Academic Editors: W.-S. Du and T. Ozawa

Copyright © 2013 Jinfeng Wang 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.

Abstract

We present a new mixed finite element method for solving the extended Fisher-Kolmogorov (EFK) equation. We first decompose the EFK equation as the two second-order equations, then deal with a second-order equation employing finite element method, and handle the other second-order equation using a new mixed finite element method. In the new mixed finite element method, the gradient belongs to the weaker space taking the place of the classical space. We prove some a priori bounds for the solution for semidiscrete scheme and derive a fully discrete mixed scheme based on a linearized Crank-Nicolson method. At the same time, we get the optimal a priori error estimates in and -norm for both the scalar unknown and the diffusion term and a priori error estimates in -norm for its gradient for both semi-discrete and fully discrete schemes.

1. Introduction

In recent years, a lot of numerical methods for solving fourth-order partial differential equations have been presented and analyzed by many researchers. In [16], authors studied some mixed finite element methods for fourth-order elliptic and parabolic equations. Shi et al. [7], Wang et al. [8, 9], and H. R. Chen and S. C. Chen [10] proposed some nonconforming finite element methods for fourth-order elliptic equation (or biharmonic equation). In [11], some rectangular finite element methods for fourth-order elliptic singular perturbation problems were considered. Hu and Shi [12] studied the best norm error estimate of lower order finite element methods for the fourth-order problem. Chen and Wang [13] discussed a mixed finite element method for thin film epitaxy. In [14], a conforming finite element approximation for the fourth-order Steklov eigenvalue problem is discussed. In [15], a Crank-Nicolson time-stepping was used to approximate the differential term and the product trapezoidal method was employed to treat the integral term, and the quasi-wavelets numerical method for space discretization. Some numerical methods were proposed and studied for Cahn-Hilliard equations, such as (mixed) finite element methods [1618], time-stepping methods [19, 20], spectral Galerkin method [21], discontinuous Galerkin method [22], and a conservative nonlinear difference scheme [23]. Liu et al. [24] studied a -conforming finite element method for a fourth-order nonlinear hyperbolic equation. He et al. [25], proposed some mixed element schemes for fourth-order linear wave equation. In [2628], some (mixed) finite element methods were studied for the extended Fisher-Kolmogorov equations.

In [29, 30], authors proposed and discussed a new mixed finite element scheme based on new mixed spaces for second-order linear elliptic equations. In the new mixed method, a weaker space was provided to replace the complex space. Considering the advantage of the new mixed method, some problems have been studied, such as second-order Sobolev equation [31], second-order parabolic equations [32, 33], and reaction-convection-diffusion problems [34, 35]. However, the new mixed element method for the extended Fisher-Kolmogorov equation has not been studied in the literatures.

In this paper, our aim is to apply the new mixed method [29, 30] to solve the extended Fisher-Kolmogorov (EFK) equation [28]

with initial condition

and Dirichlet boundary conditions

where is a bounded convex polygonal domain in , with boundary , and is the time interval with . is given function, coefficient is a positive constant.

For formulating the new mixed scheme, we first introduce a diffusion term to get the following two lower order equations:

Then, we introduce another auxiliary variable in (4b) to obtain the following lower-order system:

For the lower-order system (5a), (5b), and (5c) we will formulate a new mixed scheme based on [29, 30] and discuss some a priori bounds for the mixed element solution and a priori error estimates for semidiscrete scheme. What is more, we will get the fully discrete scheme based on linearized Crank-Nicolson method and some a priori error estimates. From the obtained results, we can find easily that a priori error estimates in and -norm for the scalar unknown and the diffusion term are optimal. Moreover, we also prove a priori error estimate in -norm for the gradient term .

The remainder of this paper is organized as follows. In Section 2, a new mixed weak formulation and semidiscrete mixed scheme are formulated for system (5a), (5b), and (5c), then, some a priori bounds for the new mixed element solution are derived. In Section 3, a new mixed elliptic projection operator and a projection operator associated with the coupled systems are presented and some a priori error estimates for semidiscrete scheme are proved. In Section 4, some a priori error estimates for fully discrete linearized Crank-Nicolson scheme are analyzed. Finally in Section 5, some concluding remarks and extensions about the new mixed finite element method are given. In this paper, is a generic constant which does not depend on the spatial mesh parameter or time step parameter . At the same time, we denote the natural inner product in or by with the corresponding norm . The others notations and definitions of Sobolev spaces as in [36, 37] are used. In order to facilitate the expression, we denote , , such as , , .

2. New Mixed Weak Formulation and Semidiscrete Scheme

In the following analysis, we will discuss the new mixed formulation and some a priori error estimates for semidiscrete scheme. Based on the lower-order system (5a), (5b), and (5c) the new mixed weak formulation is to find such that

Let and , respectively, be finite dimensional subspaces of and defined by the finite element pair [29, 30]

Then, the semidiscrete mixed scheme for (6a), (6b), and (6c) is to determine such that

In the following discussion, we derive some a priori bounds for the finite element solution.

Theorem 1. Let be the solution for system (8a), (8b), and (8c) then there exists a constant such that, for

Proof. Differentiating (8b) and (8c) with respect to , we obtain
Take , , in (10a), (10b), and (10c) to obtain
We define and add the three equations for system (11a), (11b), and (11c) to get
Integrate (12) with respect to time from to and use Gronwall lemma to have
We note that to obtain
Take in (8b) and in (8c) to obtain
Substitute (14) into (15) to obtain
Choose in (8c) and use (16) to obtain
For , we use Sobolev imbedding theorem to get
By (14), (16)–(18), we accomplish the proof of Theorem 1.

Remark 2. Based on the discussion in [29, 30], we can see easily that the discrete LBB condition for the mixed finite element space is satisfied. In our weak formulation (6a), (6b), and (6c) the space is replaced by the weaker space. Compared to the space, the regularity requirement for our space is reduced.

3. A Priori Error Estimates for Semidiscrete Scheme

In order to analyze the convergence of the method, we first introduce the projection operator and a new mixed elliptic projection operator associated with the coupled equations.

Lemma 3. Let be satisfied by the following relation:

Lemma 4. Let be given by the following new mixed relations:
there exists a constant independent of such that

from [29, 30], we can obtain the proof for Lemma 4.

Further, assuming that the triangulation is quasi-uniform, the following error estimates [27] hold, for and :

Based on Lemmas 3 and 4, the following theorem for semidiscrete error estimates is obtained.

Theorem 5. With , assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy , . Then there exists a positive constant independent of such that

Proof. To get a priori error estimates, we decompose the errors as
Combining (6a)–(8c), (19) and (20a) and (20b), we can get the error equations
Differentiating (25b) and (25c) with respect to , we get
Choose , , and in (26a), (26b), and (26c), respectively, to obtain
Using (27a), (27b), and (27c) and Cauchy-Schwarz inequality and Young inequality, we have
Using Cauchy-Schwarz inequality, Young inequality, Sobolev imbedding theorem, and inequality (22), we use the similar method as the one in [27] to obtain
Substituting (29) into (28), we have
Integrate (30) with respect to time from to to obtain
Using Gronwall lemma, we have
Taking in (25b) and in (25c), we have
Add the two equations to get
Choose in (25c) to obtain
Substitute (32) into (34) and (35) to get
Combining Lemmas 3 and 4, (32)–(36) with the triangle inequality, we obtain the error estimates for Theorem 5.

Remark 6. The conclusion for Theorem 5 demonstrates that the optimal convergence order in -norm for both the scalar unknown and the diffusion term is obtained. At the same time, the optimal convergence order in -norm for the scalar unknown is gotten, too.

Theorem 7. Assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy and . Then there exists a positive constant independent of such that

Proof. Choose , , and in (26a), (26b), and (26c), respectively, to obtain
Adding the three equations of system (38a), (38b), and (38c), we have
Integrate (39) with respect to and use Cauchy-Schwarz inequality and Young inequality to get
Using the similar method to the estimate for (29), we obtain
Substitute (41) into (40) to get
Substitute (32) into (42) and use Gronwall lemma to get
Combining Lemmas 3 and 4, (43) with the triangle inequality, we obtain the error estimates for Theorem 7.

Remark 8. In Theorem 7, the optimal convergence order in -norm for the diffusion term is obtained.

4. A Priori Error Estimates Based on Linearized C-N Scheme

4.1. Linearized Crank-Nicolson Mixed Scheme

In the following discussion, we will derive the fully discrete a priori error estimates based on a linearized Crank-Nicolson method. Let be a given partition of the time interval with step length and nodes , for some positive integer . For a smooth function on , define and .

Equations (6a), (6b), and (6c) can be written as the following formulation at time :

Then, the following equivalent formulation for (44a), (44b), and (44c) is as follows:

where

Now a linearized C-N fully discrete procedure is to find such that

Remark 9. We can find that the system (47a), (47b), and (47c) is a linear scheme by a linearized term

In the following subsection, we derive a priori error estimates based on a linearized Crank-Nicolson fully discrete scheme.

4.2. A Priori Error Estimates for Fully Discrete Scheme

In order to derive the linearized C-N fully discrete error estimates, we now write the errors as follows:

Subtracting (47a), (47b), and (47c) from (45a), (45b), and (45c) and using (19) and (20a) and (20b) at , we have the following error equations

In order to get the fully discrete error estimates, we introduce the following lemma.

Lemma 10. For and , the following estimates hold:

Proof. Using the Taylor expansion, we can obtain easily the conclusion for Lemma 10.

Remark 11. From the estimate for in Lemma 10, we know that the convergence rate in time direction for the system (47a), (47b), and (47c) is order 2, so the system (47a), (47b), and (47c) is called a linearized Crank-Nicolson scheme based on the linearized term .

In the following discussion, we will derive some fully discrete a priori error estimates.

Theorem 12. With , assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy , , , , . Then there exists a positive constant independent of and such that .

Proof. Setting , , and in (50a), (50b), and (50c), respectively, we get
Adding the three equations of (53a), (53b), and (53c) and using Cauchy-Schwarz inequality and Young inequality, we have
In order to accomplish our process, we have to estimate the nonlinear term . Using Young inequality, we have
Note that
Substitute (55) and (56) into (54) to obtain
Summing from , the resulting inequality becomes
For sufficiently small , using Gronwall lemma with Lemma 10, we get
Use Lemmas 3 and 4, (59) with the triangle inequality to accomplish the proof of Theorem 12.

Theorem 13. With , assume that the solution's regular properties for system (6a), (6b), and (6c) satisfy , , , , . Then there exists a positive constant independent of and such that

Proof. Choose , , and in (50a), (50b), and (50c) to obtain
We add the above three equations of (61a), (61b), and (61c) to get
For (62), we employ Cauchy-Schwarz inequality as well as Young inequality to obtain
Multiply (63) by and then sum from to get
Substituting the following inequality
and (55) into (64), we obtain
Substitute (59) into (66) to obtain
Taking in (50c), we have
Combining Lemmas 310, (67), and (68) with the triangle inequality, we accomplish the proof.

5. Some Concluding Remarks and Extensions

In [29, 30], authors proposed a new mixed finite element method, which has been applied to solve second-order evolution equations, such as Sobolev equations [31], parabolic equations [32, 33], and reaction-convection-diffusion problems [34, 35]. In this paper, we apply the new mixed finite element scheme [29, 30] to solve the extended Fisher-Kolmogorov (EFK) equation (fourth-order nonlinear reaction diffusion equation). Compared to the classical mixed methods, the weaker square integrable space, which takes the place of the complex , is used in our method. We derive some a priori bounds for the solution and a priori error estimates for semidiscrete scheme. What's more, we obtain a priori error estimates for fully discrete scheme by a linearized Crank-Nicolson method.

If we take in (1), we can get the following second-order nonlinear reaction-diffusion equation:

with initial condition

and Dirichlet boundary conditions

where is a bounded convex polygonal domain in , with boundary , and is the time interval with . is given function, .

Using a similar method as the one in this paper, we can get the following weak formulation for system (69)–(71):

where . Based on the mixed weak formulation (72a) and (72b), we can get the similar theoretical analysis as our method in this paper.

In the future work, we will apply the new mixed scheme to solve fourth-order linear/nonlinear wave equations [24, 25, 38]. At the same time, we will study the large time-stepping method based on the new mixed element scheme for the Cahn-Hilliard equation [19, 20].

Acknowledgments

This work was supported by the National Natural Science Fund of China (11061021), the Key Project of Chinese Ministry of Education (12024), the Scientific Research Projection of Higher Schools of Inner Mongolia (NJZZ12011, NJZY13199) and the Natural Science Fund of Inner Mongolia Province (2012MS0108, 2012MS0106, and 2011BS0102), the Program of Higher-level talents of Inner Mongolia University (125119, Z200901004, and 30105-125132), and the Scientific Research Projection of Inner Mongolia University of Finance and Economics (KY1101).

References

  1. Z. X. Chen, “Analysis of expanded mixed methods for fourth-order elliptic problems,” Numerical Methods for Partial Differential Equations, vol. 13, no. 5, pp. 483–503, 1997. View at Scopus
  2. J. C. Li, “Optimal error estimates of mixed finite element methods for a fourth-order nonlinear elliptic problem,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 183–195, 2007. View at Publisher · View at Google Scholar · View at Scopus
  3. J. C. Li, “Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions,” Advances in Computational Mathematics, vol. 23, no. 1-2, pp. 21–30, 2005. View at Publisher · View at Google Scholar · View at Scopus
  4. J. C. Li, “Optimal convergence analysis of mixed finite element methods for fourth-order elliptic and parabolic problems,” Numerical Methods for Partial Differential Equations, vol. 22, no. 4, pp. 884–896, 2006. View at Publisher · View at Google Scholar · View at Scopus
  5. X. L. Cheng, W. M. Han, and H. C. Huang, “Some mixed finite element methods for biharmonic equation,” Journal of Computational and Applied Mathematics, vol. 126, no. 1-2, pp. 91–109, 2000. View at Publisher · View at Google Scholar · View at Scopus
  6. Y. Liu, H. Li, S. He, W. Gao, and Z. C. Fang, “H1-Galerkin mixed element method and numerical simulation for the fourth-order parabolic partial differential equations,” Mathematica Numerica Sinica, vol. 34, no. 2, pp. 259–274, 2012.
  7. D. Y. Shi, S. P. Mao, and S. C. Chen, “On the anisotropic accuracy analysis of ACM's nonconforming finite element,” Journal of Computational Mathematics, vol. 23, no. 6, pp. 635–646, 2005. View at Scopus
  8. M. Wang, Z. C. Shi, and J. C. Xu, “Some n-rectangle nonconforming elements for fourth order elliptic equations,” Journal of Computational Mathematics, vol. 25, no. 4, pp. 408–420, 2007. View at Scopus
  9. W. Ming and J. C. Xu, “The Morley element for fourth order elliptic equations in any dimensions,” Numerische Mathematik, vol. 103, no. 1, pp. 155–169, 2006. View at Publisher · View at Google Scholar · View at Scopus
  10. H. R. Chen and S. C. Chen, “C0-nonconforming elements for fourth order elliptic problem,” Mathematica Numerica Sinica, vol. 35, no. 1, pp. 21–30, 2012.
  11. L. Wang, Y. K. Wu, and X. P. Xie, “Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems,” Numerical Methods for Partial Differential Equations, vol. 29, no. 3, pp. 721–737, 2013. View at Publisher · View at Google Scholar
  12. J. Hu and Z. C. Shi, “The best L2 norm error estimate of lower order finite element methods for the fourth order problem,” Journal of Computational Mathematics, vol. 30, no. 5, pp. 449–460, 2012. View at Publisher · View at Google Scholar
  13. W. B. Chen and Y. Q. Wang, “A mixed finite element method for thin film epitaxy,” Numerische Mathematik, vol. 122, no. 4, pp. 771–793, 2012. View at Publisher · View at Google Scholar
  14. H. Bi, S. X. Ren, and Y. D. Yang, “Conforming finite element approximations for a fourth-order Steklov eigenvalue problem,” Mathematical Problems in Engineering, vol. 2011, Article ID 873152, 13 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus
  15. X. H. Yang, D. Xu, and H. X. Zhang, “Crank-Nicolson/quasi-wavelets method for solving fourth order partial integro-differential equation with a weakly singular kernel,” Journal of Computational Physics, vol. 234, no. 1, pp. 317–329, 2013. View at Publisher · View at Google Scholar
  16. T. Zhang, “Finite element analysis for Cahn-Hilliard equation,” Mathematica Numerica Sinica, vol. 28, no. 3, pp. 281–292, 2006.
  17. X. B. Feng and A. Prohl, “Error analysis of a mixed finite element method for the Cahn-Hilliard equation,” Numerische Mathematik, vol. 99, no. 1, pp. 47–84, 2004. View at Publisher · View at Google Scholar · View at Scopus
  18. Q. Du, L. Ju, and L. Tian, “Finite element approximation of the Cahn-Hilliard equation on surfaces,” Computer Methods in Applied Mechanics and Engineering, vol. 200, no. 29–32, pp. 2458–2470, 2011. View at Publisher · View at Google Scholar · View at Scopus
  19. Y. N. He, Y. X. Liu, and T. Tang, “On large time-stepping methods for the Cahn-Hilliard equation,” Applied Numerical Mathematics, vol. 57, no. 5–7, pp. 616–628, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. Y. F. Yang, X. L. Feng, and Y. N. He, “Large time-stepping method based on the finite element discretization for the Cahn-Hilliard equation,” Journal of Applied Mathematics & Informatics, vol. 29, no. 5-6, pp. 1129–1141, 2011.
  21. Y. N. He and Y. X. Liu, “Stability and convergence of the spectral Galerkin method for the Cahn-Hilliard equation,” Numerical Methods for Partial Differential Equations, vol. 24, no. 6, pp. 1485–1500, 2008. View at Publisher · View at Google Scholar · View at Scopus
  22. S. M. Choo and Y. J. Lee, “A discontinuous Galerkin method for the Cahn-Hilliard equation,” Journal of Applied Mathematics and Computing, vol. 18, no. 1-2, pp. 113–126, 2005. View at Scopus
  23. S. M. Choo, S. K. Chung, and K. I. Kim, “Conservative nonlinear difference scheme for the Cahn-Hilliard equation,” Computers & Mathematics with Applications, vol. 39, no. 1-2, pp. 229–243, 2000. View at Scopus
  24. Y. Liu, H. Li, S. He, W. Gao, and Z. C. Fang, “A C1-conforming finite element method for nonlinear fourth-order hyperbolic equation,” World Academy of Science, Engineering and Technology, vol. 80, pp. 1423–1427, 2011. View at Scopus
  25. S. He, H. Li, and Y. Liu, “Analysis of mixed finite element methods for fourth-order wave equations,” Computers & Mathematics with Applications, vol. 65, no. 1, pp. 1–16, 2013. View at Publisher · View at Google Scholar
  26. L. J. T. Doss and A. P. Nandini, “An H1-Galerkin mixed finite element method for the extended Fisher-Kolmogorov equation,” International Journal of Numerical Analysis & Modeling B, vol. 3, no. 4, pp. 460–485, 2012.
  27. P. Danumjaya and A. K. Pani, “Mixed finite element methods for a fourth order reaction diffusion equation,” Numerical Methods for Partial Differential Equations, vol. 28, no. 4, pp. 1227–1251, 2012. View at Publisher · View at Google Scholar · View at Scopus
  28. P. Danumjaya and A. K. Pani, “Numerical methods for the extended Fisher-Kolmogorov (EFK) equation,” International Journal of Numerical Analysis & Modeling, vol. 3, no. 2, pp. 186–210, 2006.
  29. S. C. Chen and H. R. Chen, “New mixed element schemes for second order elliptic problem,” Mathematica Numerica Sinica, vol. 32, no. 2, pp. 213–218, 2010.
  30. F. Shi, J. P. Yu, and K. T. Li, “A new stabilized mixed finite-element method for Poisson equation based on two local Gauss integrations for linear element pair,” International Journal of Computer Mathematics, vol. 88, no. 11, pp. 2293–2305, 2011. View at Publisher · View at Google Scholar · View at Scopus
  31. D. Y. Shi and Y. D. Zhang, “High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equations,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3176–3186, 2011. View at Publisher · View at Google Scholar · View at Scopus
  32. Z. Weng, X. Feng, and P. Huang, “A new mixed finite element method based on the Crank-Nicolson scheme for the parabolic problems,” Applied Mathematical Modelling, vol. 36, no. 10, pp. 5068–5079, 2012. View at Publisher · View at Google Scholar · View at Scopus
  33. L. Li, P. Sun, and Z. D. Luo, “A new mixed finite element formulation and error estimates for parabolic problems,” Acta Mathematica Scientia, vol. 32, no. 6, pp. 1158–1165, 2012.
  34. Y. Liu, H. Li, W. Gao, S. He, and Z. C. Fang, “A novel characteristic expanded mixed method for reaction-convection-diffusion problems,” Journal of Applied Mathematics, vol. 2013, Article ID 683205, 11 pages, 2013. View at Publisher · View at Google Scholar
  35. D. Y. Shi and Q. L. Tang, “A new characteristic nonconforming mixed finite element scheme for convection-dominated diffusion problem,” Journal of Applied Mathematics, vol. 2013, Article ID 951692, 10 pages, 2013. View at Publisher · View at Google Scholar
  36. P. G. Ciarlet, Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, The Netherlands, 1978.
  37. Z. D. Luo, Mixed Finite Element Methods and Applications, Chinese Science Press, Beijing, China, 2006.
  38. S. M. Zheng and A. Milani, “Exponential attractors and inertial manifolds for singular perturbations of the Cahn-Hilliard equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 57, no. 5-6, pp. 843–877, 2004. View at Publisher · View at Google Scholar · View at Scopus