The Local Discontinuous Galerkin Finite Element Method For A Class Of Convection-Diffusion Equations

Discontinuous Galerkin method is a class of finite element methods using completelydiscontinuous piecewise polynomial space for the numerical solution and the test func-tions. It is originally devised to solve hyperbolic conservation laws containing only firstorder spatial derivatives, e.g.. It has the advantage of flexibility which is notshared by typical finite element methods, such as the allowance of arbitrary triangula-tion, even those with hanging nodes, and that the simplicity of algorithm, the facility ofparallelism. The discontinuous Glerkin (DG) method was later generalized to the localdiscontinuous Glerkin (LDG) method for solving convection-diffusion equations (contain-ing second derivatives) by Cockburn and Shu in 1998. Their work was motivated by thesuccessful numerical experiments of Bassi and Rebay for the compressible Navier-Stokesequations in 1997. The DG method has found widely applications because it is simple tocompute, easy to program and better to keep some physical properties.In this paper, we study the LDG method for a class of nonlinear convection-diffusionequations. The introduction both of one and two dimension is shown in detail.Based on the Hopf-Cole transformation, we transform the original problem into a linearheat equation with the same kind of boundary condition, the heat equation is then solvedby the LDG finite element method with special chosen numerical flux. At the same time,we give the theoretical analysis and error estimate. Theoretical analysis show that thismethod is stable and (k + 1)th order of convergence rate when the polynomials P~k areused. In addition, we present some examples both with periodic and Dirichlet boundarycondition to demonstrate the validity and the high-order accuracy of this method. Ineither cases the numerical results can achieve the optimal convergence order.