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<p>The first domain decomposition methods for partial differential equations were already developed in 1870 by H. A. Schwarz. Here we consider a nonlocal Dirichlet problem with variable coefficients, where a nonlocal diffusion operator is used. We find that domain decomposition methods like the so-called Schwarz methods seem to be a natural way to solve these nonlocal problems. In this work, we show the convergence for nonlocal problems, where specific symmetric kernels are employed, and present the implementation of the multiplicative and additive Schwarz algorithms in the above-mentioned nonlocal setting.</p>

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Schwarz Methods for Nonlocal Problems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月2日

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  • Matthias Schuster,
  • Christian Vollmann,
  • Volker Schulz

摘要

The first domain decomposition methods for partial differential equations were already developed in 1870 by H. A. Schwarz. Here we consider a nonlocal Dirichlet problem with variable coefficients, where a nonlocal diffusion operator is used. We find that domain decomposition methods like the so-called Schwarz methods seem to be a natural way to solve these nonlocal problems. In this work, we show the convergence for nonlocal problems, where specific symmetric kernels are employed, and present the implementation of the multiplicative and additive Schwarz algorithms in the above-mentioned nonlocal setting.