A collocation method for weakly singular integral equations with super-algebraic convergence rate

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• 作者：Tilo Arens ; Thomas Rösch
• 关键词：Mathematics Subject Classification65R20
• 刊名：Numerische Mathematik
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：134
• 期：3
• 页码：441-472
• 全文大小：1,202 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Mathematics
  Mathematical and Computational Physics
  Mathematical Methods in Physics
  Numerical and Computational Methods
  Applied Mathematics and Computational Methods of Engineering
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：0945-3245
• 卷排序：134

文摘

We consider biperiodic weakly singular integral equations of the second kind such as they arise in boundary integral equation methods. These are solved numerically using a collocation scheme based on trigonometric polynomials. The singularity is removed by a local change to polar coordinates. The resulting operators have smooth kernels and are discretized using the tensor product composite trapezoidal rule. We prove stability and convergence of the scheme achieving algebraic convergence of any order under appropriate regularity assumptions. The method can be applied to typical boundary value problems such as potential and scattering problems both for bounded obstacles and for periodic surfaces. We present numerical results demonstrating that the expected convergence rates can be observed in practice.