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Volume 7, Issue 1
Alternating Direction Implicit Galerkin Finite Element Method for the Two-Dimensional Time Fractional Evolution Equation

Limei Li & Da Xu

DOI: 10.4208/nmtma.2014.y11051

Numer. Math. Theor. Meth. Appl., 7 (2014), pp. 41-57.

Published online: 2014-07

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  • Abstract

New numerical techniques are presented for the solution of the two-dimensional time fractional evolution equation in the unit square. In these methods, Galerkin finite element is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) method based on the backward Euler method combined with the first order convolution quadrature approximating the integral term are considered. The ADI Galerkin finite element method is proved to be convergent in time and in the $L^2$ norm in space. The convergence order is $\mathcal{O}$($k$|ln $k$|+$h^r$), where $k$ is the temporal grid size and $h$ is spatial grid size in the $x$ and $y$ directions, respectively. Numerical results are presented to support our theoretical analysis.

  • Keywords

Fractional evolution equation, alternating direction implicit method, Galerkin finite element method, backward Euler.

  • AMS Subject Headings

65M06, 65M12, 65M15, 65M60

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-7-41, author = {Limei Li and Da Xu}, title = {Alternating Direction Implicit Galerkin Finite Element Method for the Two-Dimensional Time Fractional Evolution Equation}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2014}, volume = {7}, number = {1}, pages = {41--57}, abstract = {

New numerical techniques are presented for the solution of the two-dimensional time fractional evolution equation in the unit square. In these methods, Galerkin finite element is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) method based on the backward Euler method combined with the first order convolution quadrature approximating the integral term are considered. The ADI Galerkin finite element method is proved to be convergent in time and in the $L^2$ norm in space. The convergence order is $\mathcal{O}$($k$|ln $k$|+$h^r$), where $k$ is the temporal grid size and $h$ is spatial grid size in the $x$ and $y$ directions, respectively. Numerical results are presented to support our theoretical analysis.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2014.y11051}, url = {http://global-sci.org/intro/article_detail/nmtma/5865.html} }
TY - JOUR T1 - Alternating Direction Implicit Galerkin Finite Element Method for the Two-Dimensional Time Fractional Evolution Equation AU - Limei Li & Da Xu JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 41 EP - 57 PY - 2014 DA - 2014/07 SN - 7 DO - http://doi.org/10.4208/nmtma.2014.y11051 UR - https://global-sci.org/intro/article_detail/nmtma/5865.html KW - Fractional evolution equation, alternating direction implicit method, Galerkin finite element method, backward Euler. AB -

New numerical techniques are presented for the solution of the two-dimensional time fractional evolution equation in the unit square. In these methods, Galerkin finite element is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) method based on the backward Euler method combined with the first order convolution quadrature approximating the integral term are considered. The ADI Galerkin finite element method is proved to be convergent in time and in the $L^2$ norm in space. The convergence order is $\mathcal{O}$($k$|ln $k$|+$h^r$), where $k$ is the temporal grid size and $h$ is spatial grid size in the $x$ and $y$ directions, respectively. Numerical results are presented to support our theoretical analysis.

Limei Li and Da Xu. (2014). Alternating Direction Implicit Galerkin Finite Element Method for the Two-Dimensional Time Fractional Evolution Equation. Numerical Mathematics: Theory, Methods and Applications. 7 (1). 41-57. doi:10.4208/nmtma.2014.y11051
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