Abstract
We analyze a fully discrete leapfrog/Galerkin finite element method for the numerical solution of the space fractional order (fractional for simplicity) diffusion equation. The generalized fractional derivative spaces are defined in a bounded interval. And some related properties are further discussed for the following finite element analysis. Then the fractional diffusion equation is discretized in space by the finite element method and in time by the explicit leapfrog scheme. For the resulting fully discrete, conditionally stable scheme, we prove an L 2-error bound of finite element accuracy and of second order in time. Numerical examples are included to confirm our theoretical analysis.
Highlights
Fractional calculus and fractional partial differential equations (FPDEs) have many applications in various aspects such as in viscoelastic mechanics, power-law phenomenon in fluid and complex network, allometric scaling laws in biology and ecology, colored noise, electrode-electrolyte polarization, dielectric polarization, boundary layer effects in ducts, electromagnetic waves, quantitative finance, quantum evolution of complex systems, and fractional kinetics [1]
The finite element method has been used to find the variational solution of FPDEs [5,6,7,8,9,10,11,12,13,14]
We study the finite element method for fractional diffusion equation
Summary
Fractional calculus and fractional partial differential equations (FPDEs) have many applications in various aspects such as in viscoelastic mechanics, power-law phenomenon in fluid and complex network, allometric scaling laws in biology and ecology, colored noise, electrode-electrolyte polarization, dielectric polarization, boundary layer effects in ducts, electromagnetic waves, quantitative finance, quantum evolution of complex systems, and fractional kinetics [1]. There are still some interesting schemes that can be constructed to enhance the convergence order by using the finite difference/finite element mixed method. We use the explicit leapfrog difference/ Galerkin finite element mixed method to numerically solve the space fractional diffusion equation in order to get a higher convergence order. A more general form κ1 ⋅ RLD2aα,x + κ2 ⋅ RLD2xα,b is widely used for mathematical modelling and numerical computation. We mainly focus on constructing and analyzing a kind of efficient numerical schemes for approximately solving space fractional diffusion equation. The spatial fractional differential operator Δα is denoted by κ1 ⋅ RLD2aα,x + κ2 ⋅ RLDx2α,b, where 0 ≤ κ1, κ2 ≤ 1, and κ1 + κ2 = 1.
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