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 - <p style="text-align: justify;">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&nbsp;$\mathcal{O}$($k$|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.</p>