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.

翻译：针对偏微分方程的首个区域分解方法已于1870年由H. A. Schwarz提出。本文考虑一类具有变系数的非局部Dirichlet问题，其中采用非局部扩散算子。我们发现诸如Schwarz方法等区域分解方法似乎是求解这类非局部问题的自然途径。在本研究中，我们展示了使用特定对称核的非局部问题的收敛性，并给出了在上述非局部框架下乘性及加性Schwarz算法的实现方法。