论文标题
在bakhvalov型网格上的有限元方法平衡规范中的收敛性和超级胶丝，以解决反应 - 扩散问题
Convergence and supercloseness in a balanced norm of finite element methods on Bakhvalov-type meshes for reaction-diffusion problems
论文作者
论文摘要
在对奇异扰动反应问题的有限元方法的收敛分析中，已经成功引入了平衡的规范以替代标准能量规范，以便可以捕获层。在本文中，我们关注Bakhvalov型矩形网格的平衡规范中的收敛分析。为了实现我们的目标，引入了一个新颖的插值操作员，该操作员由局部加权$ l^2 $投影操作员和拉格朗日插值操作员组成，以用于平衡规范中最佳顺序的融合分析。该分析还取决于$ l^2 $投影的稳定性和Bakhvalov型网格的特性。此外，我们获得了平衡规范的超级知识，这是文献中首次出现的。该结果取决于另一个新颖的插入剂，该插入剂由局部加权$ l^2 $投影操作员，边界上的顶点 - 边缘元素操作员和一些校正组成。
In convergence analysis of finite element methods for singularly perturbed reaction--diffusion problems, balanced norms have been successfully introduced to replace standard energy norms so that layers can be captured. In this article, we focus on the convergence analysis in a balanced norm on Bakhvalov-type rectangular meshes. In order to achieve our goal, a novel interpolation operator, which consists of a local weighted $L^2$ projection operator and the Lagrange interpolation operator, is introduced for a convergence analysis of optimal order in the balanced norm. The analysis also depends on the stabilities of the $L^2$ projection and the characteristics of Bakhvalov-type meshes. Furthermore, we obtain a supercloseness result in the balanced norm, which appears in the literature for the first time. This result depends on another novel interpolant, which consists of the local weighted $L^2$ projection operator, a vertices-edges-element operator and some corrections on the boundary.
