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`The Scientific World JournalVolume 2013, Article ID 756281, 11 pageshttp://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 , authors studied some mixed finite element methods for fourth-order elliptic and parabolic equations. Shi et al. , Wang et al. [8, 9], and H. R. Chen and S. C. Chen  proposed some nonconforming finite element methods for fourth-order elliptic equation (or biharmonic equation). In , some rectangular finite element methods for fourth-order elliptic singular perturbation problems were considered. Hu and Shi  studied the best norm error estimate of lower order finite element methods for the fourth-order problem. Chen and Wang  discussed a mixed finite element method for thin film epitaxy. In , a conforming finite element approximation for the fourth-order Steklov eigenvalue problem is discussed. In , 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 , time-stepping methods [19, 20], spectral Galerkin method , discontinuous Galerkin method , and a conservative nonlinear difference scheme . Liu et al.  studied a -conforming finite element method for a fourth-order nonlinear hyperbolic equation. He et al. , proposed some mixed element schemes for fourth-order linear wave equation. In , 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 , 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 

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  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  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 , 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).

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