The Direct Discontinuous Galerkin Methods For Singularly Perturbed Problems

The discontinuous Galerkin method for various problems has been one of the highlights in the study of numerical methods, and has been applied to the fields of science and engineering widely in recent years. Compared with the classical DG method, the merit of the direct discontinuous Galerkin method (DDG) discussed in this paper is: instead of introducing a new variable q and adding extra boundary stability terms, only numerical trace for the derivative of the potential is used in the weak formulation. Our work will show that this approach not only can solve the model problem quite well, but also can simplify the numerical scheme and reduce the computational cost dramatically.The key of the DDG method is how to choose an appropriate numerical trace. We will introduce a strategy which not only includes the jump of U, but also the average of U_x to define at the boundary of the cell. On the other hand, for the numerical traces corresponding to the convection term, we adopt the classical upwind scheme.Based on the DDG method, many numerical experiments are performed under the uniform mesh and two-type layer-adapted meshes, i.e., the Shishkin mesh and improved grade mesh. These numerical results demonstrate that, the optimal convergence of order p+1 can be achieved if p is odd; On the other hand, the convergence of order p is attained if p is even. Fortunately, it can be modified to p+ 1 by improving the numerical traces even if p is even. More importantly, this method also leads to uniform convergence for layer-adapted meshes.