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A Two-Grid Finite Element Approximation for A Nonlinear Time-Fractional Cable Equation
In this article, a nonlinear fractional Cable equation is solved by a
two-grid algorithm combined with finite element (FE) method. A temporal
second-order fully discrete two-grid FE scheme, in which the spatial direction
is approximated by two-grid FE method and the integer and fractional
derivatives in time are discretized by second-order two-step backward
difference method and second-order weighted and shifted Gr\"unwald difference
(WSGD) scheme, is presented to solve nonlinear fractional Cable equation. The
studied algorithm in this paper mainly covers two steps: First, the numerical
solution of nonlinear FE scheme on the coarse grid is solved, Second, based on
the solution of initial iteration on the coarse grid, the linearized FE system
on the fine grid is solved by using Newton iteration. Here, the stability based
on fully discrete two-grid method is derived. Moreover, the a priori estimates
with second-order convergence rate in time is proved in detail, which is higher
than the L1-approximation result with .
Finally, the numerical results by using the two-grid method and FE method are
calculated, respectively, and the CPU-time is compared to verify our
theoretical results.Comment: 23 pages, 5 figure