Finite Element Method for the Space and Time Fractional Fokker-Planck Equation

By Weihua Deng. SIAM Journal on Numerical Analysis, 2008

582 citations recorded by the source corpus.

We develop the finite element method for the numerical resolution of the space and time fractional Fokker-Planck equation, which is an effective tool for describing a process with both traps and flights; the time fractional derivative of the equation is used to characterize the traps, and the flights are depicted by the space fractional derivative. The stability and error estimates are rigorously established, and we prove that the convergent order is $O(k^{2-\alpha}+h^\mu)$, where $k$ is the time step size and $h$ the space step size. Numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/1c0cc94f15dce527a01ad7e6436c1a1d7e97845e

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In Applied Mathematics

1. Changpin Li
2. Ian L Turner
3. Fawang Liu
4. Fanhai Zeng
5. Vo Anh
6. Da Xu
7. Wenlin Qiu
8. G. Em Karniadakis

In Modeling and Simulation

1. Changpin Li
2. Fawang Liu
3. Ian L Turner
4. Vo Anh
5. Fanhai Zeng
6. Kevin Burrage
7. Da Xu
8. Wenlin Qiu

In Numerical Analysis

1. Changpin Li
2. Fawang Liu
3. Ian L Turner
4. Vo Anh
5. Fanhai Zeng
6. Kevin Burrage
7. Da Xu
8. Wenlin Qiu

Citations in the corpus

• Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients (2015, methodology): “In the last decade or so, extensive research has been carried out on the development of numerical methods for fractional partial differential equations, including finite difference method [11–13], finite element methods [14, 15], and spectral methods [16].”
• A Crank-Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation (2014, methodology): “Up to now, there has been some work on numerical methods for FDEs, such as finite difference methods [12, 20, 26, 47], finite element methods [5, 6, 11, 14, 42], spectral methods [13, 15, 16], and so on [28, 29, 30].”
• Local discontinuous Galerkin methods for fractional diffusion equations (2013, unknown): “With the smoothness associated with solutions to fractional problems [10], the high order accuracy allows for a reduced number of degrees of freedom without impacting the accuracy.”
• A compact finite difference scheme for the fourth‐order time‐fractional integro‐differential equation with a weakly singular kernel (2019, methodology): “led to the rapid development of some numerical methods, such as finite difference methods [2, 5, 7– 16], finite element methods [17, 18], discontinuous Galerkin methods [19–21], quasi-wavelet methods [22, 23], and so on.”
• Time two-mesh algorithm combined with finite element method for time fractional water wave model (2018, background): “Email addresses: mathliuyang@imu.edu.cn; mathliuyang@aliyun.com(Y.Liu), yzd491254074@sina.com(Z.D. Yu), smslh@imu.edu.cn(H.Li), f.liu@qut.edu.au(F.W. Liu), w45j85f@163.com(J.F.Wang) Revised manuscript submitted to International Journal of Heat and Mass Transfer Dec. 2017 for symmetric space-fraction”
• Adaptive finite element method for fractional differential equations using hierarchical matrices (2016, background): “However, as the size of the problem increases, the time required to solve the final system of equations increases considerably due to the nonlocality of the fractional differential operators [43, 23, 10, 11, 30, 9, 4, 36, 37, 40, 26].”