- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 279038, 26 pages
A Fully Discrete Discontinuous Galerkin Method for Nonlinear Fractional Fokker-Planck Equation
1Department of Mathematics, Shanghai University, Shanghai 200444, China
2Department of Mathematics, Huainan Normal University, Huainan 232038, China
Received 13 July 2010; Revised 16 September 2010; Accepted 19 October 2010
Academic Editor: J. Jiang
Copyright © 2010 Yunying Zheng 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.
The fractional Fokker-Planck equation is often used to characterize anomalous diffusion. In this paper, a fully discrete approximation for the nonlinear spatial fractional Fokker-Planck equation is given, where the discontinuous Galerkin finite element approach is utilized in time domain and the Galerkin finite element approach is utilized in spatial domain. The priori error estimate is derived in detail. Numerical examples are presented which are inline with the theoretical convergence rate.
Many models in physics, chemistry are successfully described by the Langevin equation, which has been introduced almost 100 years before. And for some particular cases, say diffusion, the original Langevin equation can be transformed into the Fokker-Planck equation. Hänggi and Thomas  associated a Gaussian distribution of the increments of the noise generating process with the classical Fokker-Planck equation. Sun et al.  discussed the fractional model for anomalous diffusion. Metzler et al.  and Dubkov and Spagnolo  derived the fractional Fokker-Planck equation from different anomalous diffusion procedures. Metzler and Klafter  discussed fractional kinetic equation and its relation to the fractional Fokker-Planck equation. Dubkov et al.  introduced Fokker-Planck equation for Lévy flights. Now the Fokker-Planck equation is one of the best tools for characterizing anomalous diffusion, especially sub-/super-diffusion. Meanwhile the fractional Fokker-Planck equation has been found to be used in relatively wide field of applied sciences, such as plasma physics, population dynamics, biophysics, engineering, neuroscience, nonlinear hydrodynamics, and marking; see [7–13].
The Fokker-Planck equation describes the changes of a random function in space and in time. So different assumptions on probability density function lead to a variety of space-time equations. In this paper, we mainly study the model described by the following fractional Fokker-Planck equation, which is a special case in : where denotes left fractional derivative with order in the sense of Caputo.
There are some numerical methods to find the approximate solutions of the fractional differential equations [14–21]. But the discontinuous Galerkin finite element method is a very attractive method for partial differential equations because of its flexibility and efficiency in terms of mesh and shape functions. And the higher order of convergence can be achieved without over many iterations. Such a method was first proposed and analyzed in the early 1970s as a technique to seek numerical solutions of partial differential equations. The discontinuous Galerkin finite element method becomes a very attractive tool for the initial problems of the ODEs and the initial-boundary problems of PDEs; see [22–27].
The rest of this paper is constructed as follows. The fractional derivative space is introduced in Section 2. In Section 3, the discontinuous Galerkin finite element scheme is introduced. The existence and uniqueness of numerical solution are proved in Section 3. And the error estimate of the discontinuous Galerkin finite element approximation is studied in Section 4. Finally in Section 5, numerical examples are also taken to show the efficiency of the theoretical results.
2. Fractional Derivative Space
In this section, we firstly introduce the fractional integral, fractional Caputo derivative, and their properties.
Definition 2.1. The αth order left and right Riemann-Liouville integrals of function are defined as follows: where and .
Definition 2.2. The αth order left and right Caputo derivatives of function are defined in where and .
Definition 2.3. Let ; define a fractional derivative space as endowed with the seminorm and the norm
With the help of Fourier Transform, we can conclude that the following three expressions: , , and , are equivalent . So and can also be recognized as the seminorms of the fractional space , and when we use the seminorm of fractional space , there is no difference among them. Now we restrict our discussion in case , and the following notations are used, being rewritten as with norm or and seminorm or . We denote as the closure of under its norm.
The following lemmas are useful for our discussions later on.
Lemma 2.4 (see ). Let ; if , then
Lemma 2.5 (see ). For , , then
Lemma 2.6. For , then
Lemma 2.7 (see ). For , , then For , then For , then
Lemma 2.8. Let . The following mapping properties hold: The proof is similar to that in .
3. The Space-Time Discontinuous Galerkin Finite Element Approximation
In this section, we formulate a fully discrete discontinuous Galerkin finite element method for a type of nonlinear spacial fractional Fokker-Planck equation.
Problem 1 (nonlinear fractional Fokker-Planck equation). For , where the is a bounded domain. For positive constants and , the coefficients and satisfy
Throughout the paper, we always assume that the following mild Lipschitz continuity conditions on are satisfied: there exists a positive constant such that for , and , there have
Let be a partition of spatial domain . Define as the diameter of the element and . And let be a finite element space where is a set of polynomials with degree on a given domain . And the functions in are continuous on .
Let us consider a partition of time domain , , and , , . On each time slab , we define a discrete function space as The functions in the space can be discontinuous at the time node , but is at least left continuous and right continuous. And the functions in the space are polynomials, whose degree is no more than .
Define a discrete function space on as Moreover, the space can be verified as follows: where is a positive integer. In other words, for each the functions in are the elements of , and for each piecewise polynomial functions in of degree with possible discontinuities at the nodes . Set .
We introduce some norms on different spaces which will be used later on. And is equipped with the norms
And is equipped with the norms
In order to derive a variational form of Problem 1, we assume that is a sufficiently smooth solution of Problem 1, then multiply an arbitrary to obtain the integration formulation where denotes the inner product on . Integrating by parts in the right, noting that for all , , and the discontinuous property at the time node , one has The notation denotes the jump of the function at the time node ; that is,
Using the superscripts “” and “+” for left and right limits, respectively, the jump is described by It shows the discontinuity of the scheme. The inner product also denotes the transport process during different time-space slabs.
Thus, we define
Definition 3.1. For all , the function is a variational solution of Problem 1 provided that
Now we are ready to describe a fully discrete space-time finite element method to solve the nonlinear problem 1, where the Galerkin finite element method is used in spacial domain and discontinuous finite element approximation is used in time domain.
The value of on is replaced by the initial condition . The term including can be moved to the right-hand side of (3.16). Once it is computed, the value of on is different from that on , and it is the error introduced by the discretization in the numerical scheme.
Noting the discontinuity at each node in , the computation of can be decoupled in each time slab. Once is known on , this value is taken as an initial condition for the time slab and the following equation needs to be solved:
Next, we investigate the uniqueness and existence about the numerical scheme. First, we give the scheme in detail. For a fixed integer , let be the Lagrange polynomials associated with the abscissa ; that is,
For the quadrature in time slab , we use the Radau quadrature rule. For a given function , the following approximation holds: where . This quadrature rule is exact for all polynomials of degree . Using the linear transformation , which maps into , one gets
Then the Radau rule in can be got by
We choose as the basis functions for the piecewise polynomial function space . Then is uniquely determined by the functions , such that
Taking into (3.17), where , we have
On time slab , we define a Lagrange interpolation operator , such that where the interpolation points are Radau points. It is easy to see that for given , and are available. On time slab , , with the definition of interpolation operator, the discontinuous Galerkin scheme can be rewritten in detail as follows:
Next we introduce a lemma which is useful to prove the existence and estimate the error.
Consider a matrix , where .
It is clear that are independent of . And if , then
Lemma 3.2 (see ). Let be the matrixes If , then there holds where .
In order to prove the existence and uniqueness, we need to define a new exhibition for by where . Then is uniquely determined by the function .
We choose , where , and use the new expression to obtain the following results:
Set that and , the above equation can be rewritten as
Proof of Theorem 3.3. The vector space is a Hilbert space with finite dimension. For all , , equipped with the norm
Define a map from to itself by Since is a linear system, are all continuous maps, and the map is a continuous map from into itself. So according to the Brouwer fixed point theorem, there exists at least one fixed point such that . So (3.32) has at least one solution, denoted by . Next we investigate the uniqueness of the solution.
Let and be two solutions of (3.32), setting and summing from 1 to , then we have According to Lemma 3.2, we can see that for the first two terms of the right-hand side of (3.36), there exists a constant α, such that With the help of bounded assumption of and , we have According to the boundedness of fractional operator , inequality holds. Furthermore, we have
The fifth term of the right-hand side of (3.36) is estimated in
And by the Brouwer fixed point theorem, one gets furthermore,
By using inequalities (3.37)–(3.40) and (3.42), one has
Choosing , we have That is impossible. So the uniqueness is proved.
4. Error Estimation
Now we turn to analyze the error estimate of the D-G scheme.
Let be the approximate solution on time slab . Denote as a Ritz-Galerkin projection operator defined as follows:
And , then one has where is defined by It can be seen that is an element of .
Lemma 4.1. Let , be smooth functions on , and , , and is defined as above, then
Proof of Lemma 4.1. For all , is the solution of the following equation:
So the next equation holds
And for all , by the aid of approximation properties of and the weak form, we can derive So
Lemma 4.2. Let be defined as (4.1), then and are bounded, where .
Proof of Lemma 4.2. Using the properties of the norm, then the following inequality is valid: According to the estimate of interpolation, we get From Lemmas 2.8 and 4.1, and the inverse estimate, we know that . Therefore, it is easy to see that So The boundedness can be derived from above. The proof of can be similarly given.
Let be the usual Lagrange interpolation operator at the Radau points on ; that is, where are the Radau points. Therefore, we can see that
Lemma 4.3. The interpolation operation on has the following property:
Lemma 4.4. For defined above, we have the following error estimates:
We are now ready to prove convergence result. Putting into D-G finite element formula (3.17), and setting , we can get the basic error equation as follows: where and .
Proof of Theorem 4.5. It is necessary to rewrite and discrete error as follows:
where . We also denote as and . Noting that , , so and . Setting and ,, then the basic error equation can be rearranged as follows:
Let and sum from 1 to , then the error equation can be derived as
Consider the bounded assumption of and ; the following inequality can be got:
Since is bounded, and from Lemma 2.8, where is a linear bounded map, the following inequality is established:
The first and second terms of the left side of (4.24) can be seen as
Taking (4.24)–(4.27) into the left side of (4.22) yields
As to , for , there exists
This means that there exists constants and , which are independent of , such that
In order to estimate , we apply the interpolation operator. Let be the interpolation operator on the time slab , whose order is less than . The interpolation points involve not only Radau points, but also , and these points satisfy that . Then, for every , such that is a polynomial whose order is ,
As to , when is sufficiently small, with the aid of Hölder inequality, we obtain where
Almost similar to the estimation of , the analysis of can be got by the help of boundedness of , and .
The last term contains