The local discontinuous Galerkin finite element method for a class of convection–diffusion equations

In this paper, we study the local discontinuous Galerkin (LDG) finite element method for solving a class of convection–diffusion equations with the first-kind boundary conditions. Based on the Hopf–Cole transformation, we transform the original equation into a linear heat equation with the same kind boundary conditions. Then the heat equation is solved by the LDG finite element method with a suitably chosen numerical flux. Theoretical analysis shows that this method is stable and has a (k+1)(k+1)-th order of convergence rate when the polynomials PkPk are used. Finally, numerical experiments for one-dimensional and two-dimensional convection–diffusion equations are given to confirm the theoretical results.