论文标题
在Shishkin网状网上局部不连续的Galerkin方法平衡规范下最佳顺序的均匀收敛
Uniform convergence of optimal order under a balanced norm of a local discontinuous Galerkin method on a Shishkin mesh
论文作者
论文摘要
对于1D和2D中奇异的扰动反应扩散问题，我们在Shishkin Mesh上研究了局部不连续的Galerkin（LDG）方法。在这些情况下，标准能量规范太弱，无法充分捕获解决方案中出现的边界层的行为。为了应对这种缺陷，我们引入了比能量规范更强大的平衡规范。为了在一维情况下在平衡规范下实现最佳收敛，我们设计了新颖的数值通量，并提出了一个特殊的插值，该插值由高斯 - 拉达（Gauss-Radau）投影和局部$ l^2 $投影组成。此外，我们概括了数值通量和插值，并将最佳顺序的收敛分析从1D扩展到2D。最后，提出了数值实验以确认理论结果。
For singularly perturbed reaction-diffusion problems in 1D and 2D, we study a local discontinuous Galerkin (LDG) method on a Shishkin mesh. In these cases, the standard energy norm is too weak to capture adequately the behavior of the boundary layers that appear in the solutions. To deal with this deficiency, we introduce a balanced norm stronger than the energy norm. In order to achieve optimal convergence under the balanced norm in one-dimensional case, we design novel numerical fluxes and propose a special interpolation that consists of a Gauss-Radau projection and a local $L^2$ projection. Moreover, we generalize the numerical fluxes and interpolation, and extend convergence analysis of optimal order from 1D to 2D. Finally, numerical experiments are presented to confirm the theoretical results.
