An $$H^1$$ H 1 -Galerkin mixed finite element method for time fractional reaction–diffusion equation

Abstract

In this article, an H 1 -Galerkin mixed finite element (MFE) method for solving time fractional reaction-diffusion equation is presented. The optimal time convergence order O(Δt 2−α ) and the optimal spatial rate of convergence in H 1 and L 2 -norms for variable u and its gradient σ are derived. Moreover, some numerical results are shown to support our theoretical analysis.

A fast discontinuous finite element discretization for the space‐time fractional diffusion‐wave equation

“…For = [α, β], we assume the midpoint β+α 2 is a point of discontinuity. We can choose suitable N in partition (7) to make a grid point locate at the point of discontinuity. From [26], we know jump discontinuities belong to fractional Soblev spaces H s ( ) with s ∈ (0, 1).…”

“…The fractional models can be classified into two principal kinds: space-fractional differential equation and time-fractional one. Numerical methods and theory of solutions to problems for fractional differential equations have been studied extensively by many researchers which mainly cover finite element methods [1][2][3][4], mixed finite element methods [5][6][7], finite difference methods [8][9][10][11][12][13], finite volume (element) methods [14,15], (local) discontinuous Galerkin (L)DG methods [16], spectral methods [17,18] and so forth.…”

A space‐time spectral method for one‐dimensional time fractional convection diffusion equations

“…Another major challenge in solving TFCDEs is improving the convergence order. At an early stage, a number of researchers attempted to achieve a better convergence via the finite difference method or (discontinuous) finite element method . Wang and Jia adopted fast finite difference methods for space‐fractional diffusion equations, and they have dropped the required storage and computational cost from O ( N 2 ) and O ( N 3 ) to O ( N log N ) and O ( N ).…”

“…Yao, Sun, and Wu applied fractional alternating direction implicit method for solving a class of fractional subdiffusion equations, which led to a significant improvement in convergence. Liu developed an H 1 − Galerkin mixed finite element method for time fractional reaction–diffusion equations and obtained a convergence order of O (Δ t 2 − α ). Jin obtained an error analysis of semidiscrete finite element methods for time fractional diffusion equations (TFDEs).…”

“…Guo, Pu, and Huang [7] proved the existence and uniqueness of the fractional Ginzburg-Landau equation, the fractional QG equation, and the fractional Landau-Lifshitz equation with energy estimate, and Kaikina [8] proved the fractional evolution PDE via constructing Green's function.Another major challenge in solving TFCDEs is improving the convergence order. At an early stage, a number of researchers attempted to achieve a better convergence via the finite difference method or (discontinuous) finite element method [9][10][11][12][13][14]. Wang [9] and Jia [10] adopted fast finite difference methods for space-fractional diffusion equations, and they have dropped the required storage and computational cost from O N 2 and O N 3 to O .…”