SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 25-47, February 2024.
Abstract. Lattice Green’s functions (LGFs) are fundamental solutions to discretized linear operators, and as such they are a useful tool for solving discretized elliptic PDEs on domains that are unbounded in one or more directions. The majority of existing numerical solvers that make use of LGFs rely on a second-order discretization and operate on domains with free-space boundary conditions in all directions. Under these conditions, fast expansion methods are available that enable precomputation of 2-dimensional or 3-dimensional (3D) LGFs in linear time, avoiding the need for brute-force multidimensional quadrature of numerically unstable integrals. Here we focus on higher-order discretizations of the Laplace operator on domains with more general boundary conditions, by (1) providing an algorithm for fast and accurate evaluation of the LGFs associated with high-order dimension-split centered finite differences on unbounded domains, and (2) deriving closed-form expressions for the LGFs associated with both dimension-split and Mehrstellen discretizations on domains with one unbounded dimension. Through numerical experiments we demonstrate that these techniques provide LGF evaluations with near machine-precision accuracy, and that the resulting LGFs allow for numerically consistent solutions to high-order discretizations of the Poisson’s equation on fully or partially unbounded 3D domains.

中文翻译：

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高阶有限差分模板的 Lattice Green 函数

SIAM 数值分析杂志，第 62 卷，第 1 期，第 25-47 页，2024 年 2 月。
摘要。格格林函数 (LGF) 是离散线性算子的基本解，因此它们是在一个或多个方向上无界的域上求解离散椭圆偏微分方程的有用工具。大多数利用 LGF 的现有数值求解器都依赖于二阶离散化，并在所有方向上具有自由空间边界条件的域上运行。在这些条件下，可以使用快速展开方法，在线性时间内预先计算 2 维或 3 维 (3D) LGF，从而避免了对数值不稳定积分进行强力多维求积的需要。在这里，我们重点关注拉普拉斯算子在具有更一般边界条件的域上的高阶离散化，方法是（1）提供一种算法，用于快速准确地评估与无界域上的高阶维数分裂中心有限差分相关的 LGF， (2) 导出与具有一无界维度的域上的维度分割和 Mehrstellen 离散化相关的 LGF 的闭合形式表达式。通过数值实验，我们证明这些技术提供了接近机器精度的 LGF 评估，并且所得的 LGF 允许在完全或部分无界 3D 域上对泊松方程的高阶离散化提供数值一致的解。