Alternating direction implicit-spectral element method (ADI-SEM) for solving multi-dimensional generalized modified anomalous sub-diffusion equation

Abstract The main aim of the current paper is to solve the multi-dimensional generalized modified anomalous sub-diffusion equation by using a new spectral element method. At first, the time variable has been discretized by a finite difference scheme with second-order accuracy. The stability and convergence of the time-discrete scheme have been investigated. We show that the time-discrete scheme is unconditionally stable and the convergence order is O ( τ 2 ) in the temporal direction. Secondly, the Galerkin spectral element method has been combined with alternating direction implicit idea to discrete the space variable. The unconditional stability and convergence of the full-discrete scheme have been proved. By developing the proposed scheme, we need to calculate one-dimensional integrals for two-dimensional problems and two-dimensional integrals for three-dimensional problems. Thus, the used CPU time for the presented numerical procedure is lower than the two- and three-dimensional Galerkin spectral element methods. Also, the proposed method is suitable for computational domains obtained from the tensor product. Finally, two examples are analyzed to check the theoretical results.

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