Purpose

This paper aims to present a meshless technique to find the Green’s functions for solutions of Laplacian boundary value problems on rectangular domains. This paper also investigates a theoretical basis for the Steklov series expansion methods to reduce and estimate the error of numerical approaches for the boundary correction kernel of the Laplace operator.

Design/methodology/approach

The main interest is how the Green's functions differ from the fundamental solution of the Laplace operator. Steklov expansion methods for finding the correction term are supported by the analysis that bases of the class of all finite harmonic functions can be formed using harmonic Steklov eigenfunctions. These functions construct a basis of the space of solutions of harmonic boundary value problems and their boundary traces generate an orthogonal basis of the trace space of solutions on the boundary.

Findings

The main conclusion is that the boundary correction term for the Green's functions is well-approximated by Steklov expansions with a few Steklov eigenfunctions. The error estimates for the Steklov approximations of the boundary correction term involved in Dirichlet or Robin boundary value problems are found. They appear to provide very good approximations in the interior of the region and become quite oscillatory close to the boundary.

Originality/value

This paper concentrates to document the first attempt to find the Green's function for various harmonic boundary value problems with the explicit Steklov eigenfunctions without concerns regarding discretizations when the region is a rectangle.

中文翻译：

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拉普拉斯方程式格林函数的Steklov逼近

目的

本文旨在提出一种无网格技术，以找到格林函数来求解矩形域上的拉普拉斯边值问题。本文还研究了Steklov级数展开方法以减少和估计Laplace算子的边界校正内核的数值方法的误差的理论基础。

设计/方法/方法

主要的兴趣是格林函数与拉普拉斯算子的基本解决方案有何不同。分析可以支持使用Steklov展开方法来找到校正项，因为可以使用谐波Steklov特征函数形成所有有限谐波函数类别的基础。这些函数构成了谐波边值问题解空间的基础，并且它们的边界轨迹在边界上生成了解的轨迹空间的正交基础。

发现

主要结论是，格林函数的边界校正项与带有几个Steklov特征函数的Steklov展开很好地近似。找到Dirichlet或Robin边值问题所涉及的边界校正项的Steklov近似的误差估计。它们似乎在区域内部提供了很好的近似值，并且在边界附近变得非常振荡。

创意/价值

本文集中于记录首次尝试寻找具有明确Steklov特征函数的各种谐波边值问题的格林函数，而无需担心该区域为矩形时的离散化问题。