On Galerkin spectral element method for solving Riesz fractional diffusion equation based on Legendre polynomials

This paper presents a Galerkin spectral element method for solving a fractional diffusion equation, considering initial and boundary conditions. We construct a discrete scheme for time, employing the Crank-Nicolson method to approximate the Caputo fractional derivative on a uniform mesh. Then we introduce a Galerkin variational formulation to establish the unconditional stability of the scheme. Moreover, we apply the spectral element method based on Legendre polynomials in the space direction and obtain the fully discrete scheme. The error analysis of the fully discrete scheme is treated in L_۲ sense. we present a computational analysis to deal with the Galerkin spectral element method, to compute the corresponding bilinear form, on the implementation process. Finally, we prove the effectiveness of the method through numerical experiments and some simulations using MATLAB software.

کلیدواژه ها:

Fractional diffusion equation (FDE) ، Riesz derivative ، Caputo derivative ، ، Galerkin spectral element method ، Legendre polynomials ، stability ، error estimates