DG Method For Singularly Perturbed Problems Using Graded Meshes

In recent years, the DG method for singularly perturbed problems hasbeen one of the highlights in the study of numerical methods. The uniformconvergence and the uniform superconvergence of the DG solution have beenobserved under Shishkin meshes by some researchers.However, the theoreticalanalysis is a challenging problem. In this paper, we first prove the uniform con-vergence of p-order projection errors under one-dimensional Shishkin meshesand the improvedλ-graded meshes, i.e.,‖u-Ï€^±u‖≤Ch^p+1. Then the p-order projection errors‖D^α(u-Ï€^±u)‖≤Ch^p+1-α,‖D^α(q-Ï€^±q)‖≤Ch^p+1-α,a=0, 1, in the exterior domain of the boundary layer are obtained. Theseestimates above will play an important role condition in the verification of theDG solution's uniform convergence.Further, we provide the numerical results of DG method for one-dimensionaland two-dimensional singularly perturbed problems under the two improvedλ-graded meshes. The numerical results indicate that the improvedλ-gradedmeshes not only keep the advantages of the Shishkin meshes, but is also moreefficient and stable than the Shishkin meshes. Especially, the factor lnN iseliminated under the firstλ-graded meshes. It is more exciting that we ob-tain the derivate's uniform superconvergence estimates‖(?)_u‖_∞≤CN^-(p+2),‖q-(?)‖_∞≤CN^-(p+2)|u|_1,∞ under the two types of improvedλ-graded meshes.On the other hand, the L² error estimates under these improved meshes,‖u-U‖_L²â‰¤CN^-(p+1) is obtained.