Local Discontinuous Galerkin Finite Element Methods For Allen-Cahn Equation

The discontinuous finite element method is a numerical method which developedfrom the high-resolution finite difference method and the finite volume method and has the advantages of them. Since the end of 1980's, it attracted the attention of mathematicians and was developed well. In this paper, we develop a local discontinuousGalerkin finite element method for solving the Allen-Cahn equation, and prove L² stability and error estimates. At last, numerical examples are given.This article is divided into four parts.The first part is an introduction. In this part we briefly introduce the origin and development of the local discontinuous Galerkin finite element method and make a summary of the advantages of this method. The second part introduces variational formulation and space discrete scheme. We consider the following Allen-Cahn equationWe let q=u_x, and find q_h(z,t),u_h(x,t)∈U_h, such that, for all the test functions ω_h(x),v_h(x)∈V_h^k,where u_h,0(x) is a approximation of u₀(x) . Then we make an explanation of the time discretization used the method of Runge-Kutta Shu raised. This method maintains the stability and high-precision. We prove L² stability for the time discretization by choosing appropriate flux,We use L²-projection to get the error estimates,The third part gives numerical examples applying the local discontinuous Galerkin finite element method. And the last part is a conclusion of the paper.