AI 中文总结
研究一维局部与非局部扩散问题的耦合方法，受区域分解法启发，利用局部边界条件，构建局部到非局部的耦合方法，经有限元法等离散化，通过数值实验验证其具有可量化的$O(h)$收敛性。
AI 中文摘要
我们提出了一种用于一维局部和非局部扩散问题的新型耦合方法。与其他方法不同，我们的耦合方法仅使用局部边界条件，这是因为我们的非局部算子通过构造来强制实现。利用这一优势特性，我们构建了一种非常自然的无缝耦合。局部边界条件的使用使得成熟的数值方法能够从局部问题转移到非局部问题。我们的局部到非局部耦合方法受区域分解方法启发，旨在将其转移到非局部问题。我们研究的主要结果是构建了一种局部到非局部耦合方法，其具有可量化的$O(h)$收敛性，对任意解都成立。分别采用有限元法和伽辽金投影对局部和非局部问题进行离散化，并通过大量数值实验验证了收敛速度。
英文摘要
We present a novel coupling method for local and nonlocal diffusion problems in 1D. Unlike other methods, our coupling method exclusively uses local boundary conditions. This is possible because our nonlocal operators enforce them by construction. Leveraging this advantageous property, we construct a seamless coupling that is remarkably natural. The utilization of local boundary conditions allows for the transfer of well-established numerical methods from local problems to nonlocal ones. Our local-to-nonlocal coupling method is inspired by the domain decomposition method, which we would like to transfer to nonlocal problems. The main result of our study is the construction of a local-to-nonlocal coupling method with a quantifiable $O(h)$ convergence that holds for an arbitrary solution. For discretization of the local and nonlocal problems, the finite element method and the Galerkin projection are employed, respectively. We verify our convergence rate with extensive numerical experiments.
