An Accelerated Method for Nonlinear Elliptic PDE

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• 作者：Hayden Schaeffer ; Thomas Y. Hou
• 关键词：Nonlinear elliptic PDE ; Degenerate elliptic PDE ; Accelerated convergence ; Elliptic systems ; Finite difference methods ; Viscosity solutions ; Fixed point methods
• 刊名：Journal of Scientific Computing
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：69
• 期：2
• 页码：556-580
• 全文大小：3,115 KB
• 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
• 卷排序：69

文摘

We propose two numerical methods for accelerating the convergence of the standard fixed point method associated with a nonlinear and/or degenerate elliptic partial differential equation. The first method is linearly stable, while the second is provably convergent in the viscosity solution sense. In practice, the methods converge at a nearly linear complexity in terms of the number of iterations required for convergence. The methods are easy to implement and do not require the construction or approximation of the Jacobian. Numerical examples are shown for Bellman’s equation, Isaacs’ equation, Pucci’s equations, the Monge–Ampère equation, a variant of the infinity Laplacian, and a system of nonlinear equations.