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...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.展开更多
In this paper, a new DG method was designed to solve the model problem of the one-dimensional singularly-perturbed convection-diffusion equation. With some special chosen numerical traces, the existence and uniqueness...In this paper, a new DG method was designed to solve the model problem of the one-dimensional singularly-perturbed convection-diffusion equation. With some special chosen numerical traces, the existence and uniqueness of the DG solution is provided. The superconvergent points inside each element are observed. Particularly, the 2p ＋ 1-order superconvergence and even uniform superconvergence under layer-adapted mesh are observed numerically.展开更多
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
文摘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the National Natural Science Foundation of China(No.10571053)Programme for New Century Excellent Talents in University(NCET-06-0712)the Excellent Youth Project of the Education Department of Hunan Province of China(0513039)
文摘In this paper, a new DG method was designed to solve the model problem of the one-dimensional singularly-perturbed convection-diffusion equation. With some special chosen numerical traces, the existence and uniqueness of the DG solution is provided. The superconvergent points inside each element are observed. Particularly, the 2p ＋ 1-order superconvergence and even uniform superconvergence under layer-adapted mesh are observed numerically.
