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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

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

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:

The rest of this paper is constructed as follows. In Section

In this section, we firstly introduce the fractional integral (or Riemann-Liouville integral), the Caputo fractional derivative, and their corresponding fractional derivative space.

The

The

If

The fractional derivative space

Let

Define the seminorm

Let

The fractional space

The following are some useful results.

For

For

Since

Let

We consider equations of the form

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

In order to derive a variational form of Problem

Rewriting the above expression yields

We define the associated bilinear form

For given

A function

Now we are ready to describe a fully discrete Galerkin finite element method to solve nonlinear Problem

For a positive integer

Let

The linear systems in the above equation requires selecting the value of

For

With the assumption of

For

Let

As scheme represents a finite system of problem, the continuity and coercivity of

Let

Let

Using the definition of

So we have

Let

Let

The following norms are also used in the analysis:

Assume that Problem

For

Subtracting the above equation from the fully discrete scheme (

Substituting (

Secondly, we deduce the estimation of

The

Setting

Letting

Using

Hence, using the interpolation property and

Also using the interpolation property, Gronwall’s lemma, and the approximation properties, we get

In this section, we present the numerical results which confirm the theoretical analysis in Section

Let

The following equation

If we select

Table

Numerical error result for Example

cvge. rate | cvge. rate | |||
---|---|---|---|---|

1/5 | 2.2216 | — | 1.0213 | — |

1/10 | 1.3551 | 0.7132 | 6.0779 | 0.74875 |

1/20 | 5.5865 | 1.2784 | 2.3188 | 1.3901 |

1/40 | 3.0515 | 0.8724 | 1.0545 | 1.1367 |

1/80 | 1.2423 | 1.2964 | 3.9883 | 1.4027 |

1/160 | 5.1033 | 1.2835 | 2.1310 | 0.9042 |

The function

If we select

Table

Numerical error result for Example

cvge. rate | cvge. rate | |||
---|---|---|---|---|

1/5 | 1.3010 | — | 3.2223 | — |

1/10 | 4.6402 | 1.4878 | 1.4133 | 1.1890 |

1/20 | 1.6843 | 1.4620 | 6.2946 | 1.1669 |

1/40 | 6.6019 | 1.6843 | 2.8571 | 1.1395 |

1/80 | 2.7979 | 1.2386 | 1.3137 | 1.1209 |

1/160 | 1.2665 | 1.1434 | 6.0848 | 1.1103 |

Consider the following space-fractional differential equation with the nonhomogeneous boundary conditions,

We still choose

The numerical results are presented in Table

Numerical error result for Example

cvge. rate | cvge. rate | |||
---|---|---|---|---|

1/5 | 8.3052 | — | 2.8009 | — |

1/10 | 3.6038 | 1.2045 | 1.0086 | 1.4735 |

1/20 | 1.3839 | 1.3807 | 3.2327 | 1.6414 |

1/40 | 5.0631 | 1.4507 | 1.1789 | 1.4554 |

1/80 | 1.8920 | 1.4201 | 5.9555 | 0.9851 |

1/160 | 9.9899 | 0.9214 | 3.0034 | 0.9876 |

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.