Pointwise Error Estimates and Two-Grid Algorithms of Discontinuous Galerkin Method for Strongly Nonlinear Elliptic Problems

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• 作者：Chunjia Bi ; Cheng Wang ; Yanping Lin
• 关键词：Pointwise error estimates ; Two ; grid algorithms ; Discontinuous Galerkin methods ; Nonlinear problems
• 刊名：Journal of Scientific Computing
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：67
• 期：1
• 页码：153-175
• 全文大小：592 KB
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• 作者单位：Chunjia Bi (1)
  Cheng Wang (2)
  Yanping Lin (3)
  
  1. Department of Mathematics, Yantai University, Yantai, Shandong, 264005, People’s Republic of China
  2. Department of Mathematics, Tongji University, Shanghai, 200092, People’s Republic of China
  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, People’s Republic of China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691

文摘

In this paper, we consider the discontinuous Galerkin finite element method for the strongly nonlinear elliptic boundary value problems in a convex polygonal \( \varOmega \subset {\mathbb R}^2.\) Optimal and suboptimal order pointwise error estimates in the \(W^{1,\infty }\)-seminorm and in the \(L^{\infty }\)-norm are established on a shape-regular grid under the regularity assumptions \(u\in W^{r+1,\infty }(\varOmega ), r\ge 2\). Moreover, we propose some two-grid algorithms for the discontinuous Galerkin method which can be thought of as some type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a nonlinear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the nonlinear elliptic problem on a coarser space. Convergence estimates in a mesh-dependent energy norm are derived to justify the efficiency of the proposed two-grid algorithms. Numerical experiments are also provided to confirm our theoretical findings.