摘要
The convergence and superconvergence properties of the discontinuous Galerkin （DG） method for a singularly perturbed model problem in one-dimensional setting are studied. By applying the DG method with appropriately chosen numerical traces, the existence and uniqueness of the DG solution, the optimal order L2 error bounds, and 2p＋ 1-order superconvergence of the numerical traces are established. The numerical results indicate that the DG method does not produce any oscillation even under the uniform mesh. Numerical experiments demonstrate that, under the uniform mesh, it seems impossible to obtain the uniform superconvergence of the numerical traces. Nevertheless, thanks to the implementation of the so-called Shishkin-type mesh, the uniform 2p ＋ 1-order superconvergence is observed numerically.
The convergence and superconvergence properties of the discontinuous Galerkin （DG） method for a singularly perturbed model problem in one-dimensional setting are studied. By applying the DG method with appropriately chosen numerical traces, the existence and uniqueness of the DG solution, the optimal order L2 error bounds, and 2p＋ 1-order superconvergence of the numerical traces are established. The numerical results indicate that the DG method does not produce any oscillation even under the uniform mesh. Numerical experiments demonstrate that, under the uniform mesh, it seems impossible to obtain the uniform superconvergence of the numerical traces. Nevertheless, thanks to the implementation of the so-called Shishkin-type mesh, the uniform 2p ＋ 1-order superconvergence is observed numerically.
基金
This work is supported in part by the National Natural Science Foundation of China （10571053）
Program for New Century Excellent Talents in University, and the Scientific Research Fund of Hunan Provincial Education Department （0513039）
The second author is supported in part by the US National Science Foundation under grants DMS-0311807 and DMS-0612908
