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Yunying Zheng, Zhengang Zhao, "A Fully Discrete Galerkin Method for a Nonlinear Space-Fractional Diffusion Equation", Mathematical Problems in Engineering, vol. 2011, Article ID 171620, 20 pages, 2011. https://doi.org/10.1155/2011/171620
A Fully Discrete Galerkin Method for a Nonlinear Space-Fractional Diffusion Equation
The spatial transport process in fractal media is generally anomalous. The space-fractional advection-diffusion equation can be used to characterize such a process. In this paper, a fully discrete scheme is given for a type of nonlinear space-fractional anomalous advection-diffusion equation. In the spatial direction, we use the finite element method, and in the temporal direction, we use the modified Crank-Nicolson approximation. Here the fractional derivative indicates the Caputo derivative. The error estimate for the fully discrete scheme is derived. And the numerical examples are also included which are in line with the theoretical analysis.
The normal diffusive motion is modeled to describe the standard Brownian motion. The relation between the flow and the divergence of the particle displacement represents where is the diffusive flow. Inserting the above equation into the equation of mass conservation we obtain the standard convection-diffusion equation. From the viewpoint of physics, it means that during the method of time random walkers, the overall particle displacement up to time can be represented as a sum of independent random steps, in the case that both the mean-squared displacement per step and the mean time needed to perform a step are finite. The measured variance growth in the direction of flow of tracer plumes is typically at a Fickian rate, .
The transport process in fractal media cannot be described with the normal diffusion. The process is nonlocal and it does not follow the classical Fickian law. It depicts a particle in spreading tracer cloud which has a standard deviation, and which grows like for some , excluding the Fickian case . The description of anomalous diffusion means that the measure variance growth in the direction of flow has a deviation from the Fickian case, it follows the super-Fickian rate when , or does the subdiffusion rate if . With the help of the continuous time random walk and the Fourier transform, the governing equation with space fractional derivative can be derived as follows where denotes integer derivative respect to , and is fractional derivative. There are some authors studying the spacial anomalous diffusion equation in theoretical analysis and numerical simulations [1–10]. Now the fractional anomalous diffusion becomes a hot topic because of its widely applications in the evolution of various dynamical systems under the influence of stochastic forces. For example, it is a well-suited tool for the description of anomalous transport processes in both absence and presence of external velocities or force fields. Since the groundwater velocities span many orders of magnitude and give rise to diffusion-like dispersion (a term that combines molecular diffusion and hydrodynamic dispersion), the fractional diffusion is an important process in hydrogeology. It can be used to describe the systems with reactions and diffusions across a wide range of applications including nerve cell signaling, animal coat patterns, population dispersal, and chemical waves. In general, fractional anomalous diffusions have numerous applications in statistical physics, biophysics, chemistry, hydrogeology, and biology [4, 11–20].
In this paper, we mainly study one kind of typical nonlinear space-fractional partial differential equations by using the finite element method, which reads in the following form: where is a spacial domain with boundary , is the th order fractional derivative with respect to the space variable in the Caputo sense (which will be introduced later on), are functions of and are known functions which satisfy the conditions requested by the theorem of error estimations.
The rest of this paper is constructed as follows. In Section 2 the fractional integral, fractional derivative, and the fractional derivative spaces are introduced. The error estimates of the finite element approximation for (1.4) are studied in Section 3, and in Section 4, numerical examples are taken to verify the theoretical results derived in Section 3.
2. Fractional Derivative Space
In this section, we firstly introduce the fractional integral (or Riemann-Liouville integral), the Caputo fractional derivative, and their corresponding fractional derivative space.
Definition 2.1. The th order left and right Riemann-Liouville integrals of function are defined as follows where , and is the Gamma function.
Definition 2.2. The th order Caputo derivative of function is defined as, The th order Riemann-Liouville derivative of function is defined by changing the order of integration and differentiation.
Lemma 2.3 (see ). If , then the Caputo fractional derivative is equal to the Riemann-Liouville derivative.
Definition 2.4. The fractional derivative space is defined as follows: endowed with the seminorm and the norm
Let denote the closure of with respect to the above norm and seminorm.
Definition 2.5. Define the seminorm and the norm where is the imaginary unit, and is the Fourier transform, and which can define another fractional derivative space .
Let denote the closure of with respect to the norm and seminorm.
Definition 2.6. The fractional space is defined below endowed with the seminorm and the norm
Lemma 2.8 (see ). For , , then
Lemma 2.9 (see ). For , one has For ,
Since , , and are equal with equivalent seminorm and norm, the norms with each space which will be used following are without distinction, and the notations are used seminorm and norm .
3. Finite Element Approximation
Let , and . Define . In this section, we will formulate a fully discrete Galerkin finite element method for a type of nonlinear anomalous diffusion equation as follows.
Problem 1 (Nonlinear spacial anomalous diffusion equation). We consider equations of the form We always assume that
The algorithm and analysis in this paper are applicable for a large class of linear and nonlinear functions (including polynomials and exponentials) in the unknown variables. Throughout the paper, we assume the following mild Lipschitz continuity conditions on , and : there exist positive constants and such that for , and ,
Rewriting the above expression yields
We define the associated bilinear form as where denotes the inner product on and .
For given , we define the associated function as
Definition 3.1. A function is a variational solution of Problem 1 provided that
Now we are ready to describe a fully discrete Galerkin finite element method to solve nonlinear Problem 1. In our new scheme, the finite element trial and test spaces for Problem 1 are chosen to be same.
For a positive integer , let be a uniform partition of the time interval such that , where , and let . Throughout the paper, we use the following notation for a function :
Let be a partition of spatial domain . Define as the diameter of the element and . And let be a finite element space where is the set of polynomials of degree on a given domain . And the functions in are continuous on . Our fully discrete quadrature scheme to solve Problem 1 is to find : for such that
The linear systems in the above equation requires selecting the value of and . Given depending on the initial data , we select by solving the following predictor-corrector linear systems:
Lemma 3.2. For , there exist constants such that
Proof. With the assumption of in (3.3) and the property of dual space
Lemma 3.3 (see ). For , one has where .
Theorem 3.4. Let be bounded, then for a sufficiently small step , there exists a unique solution satisfying scheme (3.13).
Proof. As scheme represents a finite system of problem, the continuity and coercivity of is the sufficient and essential condition for the existence and uniqueness of . Let , then
For the chosen sufficiently small , the above inequality holds.
Hence, the scheme (3.13) is uniquely solvable for .
Let , and , then where is a Rits-Galerkin projection operator defined as follows:
Lemma 3.5. Let be smooth functions on , , and is defined as above, then
Proof. Using the definition of , one gets
where . Utilizing the interpolation of leads to
Next we estimate . For , is the solution of the following equation:
So we have For , with the help of approximation properties of and the weak form, we can obtain
Lemma 3.6 (see ). Let , denote a quasiuniform family of subdivisions of a polyhedral domain . Let be a reference finite element such that is a finite-dimensional space of functions on is a basis for , where , and . For , let be the affine equivalent element, and is measurable and . Then there exists a constant such that The following Gronwall’s lemma is useful for the error analysis later on.
Lemma 3.7 (see ). Let and (for integer be nonnegative numbers such that
for . Suppose that , for all , and set . Then
The following norms are also used in the analysis:
Theorem 3.8. Assume that Problem 1 has a solution satisfying with . If , then the finite element approximation is convergent to the solution of Problem 1 on the interval (0,T], as . The approximation also satisfies the following error estimates
Proof. For , find such that
Subtracting the above equation from the fully discrete scheme (3.13), and substituting into it, we obtain the following error formulation relating to and : Setting , we obtain Note that According to (3.2) and Lemma 3.2, we have From Lemma 3.3, the following inequality can be derived:
Substituting (3.37)–(3.39) into (3.36) then multiplying (3.36) by , summing from to , we have We now estimate to in the right hand of (3.40),
Secondly, we deduce the estimation of , where The estimations of and can be derived as follows: Thirdly, it is turn to consider , Next, where Rewriting by the aid of (3.20), we have The estimation of is deduced as follows: Last, we estimate , where
The should be estimated with (3.14). Let then subtracting (3.34) from the two equations of (3.14), respectively, one gets
Setting , and using the similar estimation (see (3.40)), one has
Letting , applying the above result of , and using the similar estimation (see (3.53)), we get
Using and Gronwall’s lemma, we get
Hence, using the interpolation property and the estimate (3.32) holds.
Also using the interpolation property, Gronwall’s lemma, and the approximation properties, we get which is just the estimate (3.33).
4. Numerical Examples
In this section, we present the numerical results which confirm the theoretical analysis in Section 3.
Let denote a uniform partition on , and the space of continuous piecewise linear functions on , that is, .. In order to implement the Galerkin finite element approximation, we adapt finite element discrete along the space axis, and finite difference scheme along the time axis. We associate shape function of space with the standard basis of hat functions on the uniform grid of size . We have the predicted rates of convergence if the condition of provided that the initial value is smooth enough.
Example 4.1. The following equation
has a unique solution .
If we select and note that the initial value is smooth enough, then we have
Table 1 includes numerical calculations over a regular partition of . We can observe the experimental rates of convergence agree with the theoretical rates for the numerical solution.
Example 4.2. The function solves the equation in the following form:
If we select , then
Table 2 shows the error results at different size of space grid. We can observe that the experimental rates of convergence still support the theoretical rates.
Example 4.3. Consider the following space-fractional differential equation with the nonhomogeneous boundary conditions,
whose exact solution is .
We still choose , then get the convergence rates
The numerical results are presented in Table 3 which are in line with the theoretical analysis.
In this paper, we propose a fully discrete Galerkin finite element method to solve a type of fractional advection-diffusion equation numerically. In the temporal direction we use the modified Crank-Nicolson method, and in the spatial direction we use the finite element method. The error analysis is derived on the basis of fractional derivative space. The numerical results agree with the theoretical error estimates, demonstrating that our algorithm is feasible.
This work was partially supported by the National Natural Science Foundation of China under grant no. 10872119, the Key Disciplines of Shanghai Municipality under grant no. S30104, the Key Program of Shanghai Municipal Education Commission under grant no. 12ZZ084, and the Natural Science Foundation of Anhui province KJ2010B442.
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Copyright © 2011 Yunying Zheng and Zhengang Zhao. 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.