Volume 4, Issue 4
Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation

Na Zhang, Weihua Deng & Yujiang Wu

Adv. Appl. Math. Mech., 4 (2012), pp. 496-518.

Published online: 2012-04

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  • Abstract

We present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on the $L1$-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.

  • Keywords

Modified subdiffusion equation, finite difference method, finite element method, stability, convergence rate.

  • AMS Subject Headings

26A33, 65M06, 65M12, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-4-496, author = {}, title = {Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2012}, volume = {4}, number = {4}, pages = {496--518}, abstract = {

We present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on the $L1$-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.10-m1210}, url = {http://global-sci.org/intro/article_detail/aamm/133.html} }
TY - JOUR T1 - Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 496 EP - 518 PY - 2012 DA - 2012/04 SN - 4 DO - http://doi.org/10.4208/aamm.10-m1210 UR - https://global-sci.org/intro/article_detail/aamm/133.html KW - Modified subdiffusion equation, finite difference method, finite element method, stability, convergence rate. AB -

We present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on the $L1$-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.

Na Zhang, Weihua Deng & Yujiang Wu. (1970). Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation. Advances in Applied Mathematics and Mechanics. 4 (4). 496-518. doi:10.4208/aamm.10-m1210
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