ADI Finite Element Method for 2D Nonlinear Time Fractional Reaction-Subdiffusion Equation

Peng Zhu,Shenglan Xie

doi:10.4236/ajcm.2016.64034

Peng Zhu, Shenglan Xie

Open Access

https://doi.org/10.4236/ajcm.2016.64034

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Abstract

In this paper, an alternating direction Galerkin finite element method is presented for solving 2D time fractional reaction sub-diffusion equation with nonlinear source term. Firstly, one order implicit-explicit method is used for time discretization, then Galerkin finite element method is adopted for spatial discretization and obtain a fully discrete linear system. Secondly, Galerkin alternating direction procedure for the system is derived by adding an extra term. Finally, the stability and convergence of the method are analyzed rigorously. Numerical results confirm the accuracy and efficiency of the proposed method.

Highlights

In this paper, we consider the following two-dimensional nonlinear fractional reactionsubdiffusion equation ∂u= ( x,t ) ∂t (1)( x,t ) ∈ ΩT ≡ Ω × (0,T ], with boundary and initial conditions= u ( x,t ) 0 ( x,t ) ∈ ∂Ω × (0,T ], = u ( x, 0) φ ( x) x ∈ Ω, (2)where 0 < α < 1, A = diag (κ1,κ2 ), φ ( x) is sufficiently smooth function
An alternating direction Galerkin finite element method is presented for solving 2D time fractional reaction sub-diffusion equation with nonlinear source term
Since the first Alternating direction implicit (ADI) based finite difference (FD) scheme presented for 2D space fractional diffusion equation by Meerschaert, Scheffler and Tadjeran [9], there are many literatures about various multidimensional fractional differential equations numerically solved by ADI technique

Summary

Introduction

We consider the following two-dimensional nonlinear fractional reactionsubdiffusion equation. Derived an implicit compact finite difference scheme for solving 2D nonlinear fractional reaction-subdiffusion equation. Since the first ADI based finite difference (FD) scheme presented for 2D space fractional diffusion equation by Meerschaert, Scheffler and Tadjeran [9], there are many literatures about various multidimensional fractional differential equations numerically solved by ADI technique. Zhang and Sun [17] proposed a Crank-Nicolson compact ADI FD scheme for Equation (6), where stability and two error estimates are proved rigorously by energy method. Many realistic problems involve nonlinear fractional differential equations Based on these motivations, our attention in this paper will focus on developing ADI FE schemes for efficiently solving a class of nonlinear time fractional differential equations. C and Ci denote generic positive constants independent of τ , N and n , and their value will not be the same in different equations or inequalities

Preliminary and Notations

Formulation of ADI FEM

Stability and Error Estimate

Matrix Form of ADI FEM

Numerical Experiments

Conclusion