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On the choice of source points in the method of fundamental solutions

机译：基本解法中源点的选择

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摘要

The method of fundamental solutions (MFS) may be seen as one of the simplest methods for solving boundary value problems for some linear partial differential equations (PDEs). It is a meshfree method that may present remarkable results with a small computational effort. The meshfree feature is particularly attractive when we need to change the shape of the domain, which occurs, for instance, in shape optimization and inverse problems. The MFS may be viewed as a Trefftz method, where the approximations have the advantage of verifying the linear PDE, and therefore we may bound the inner error from the boundary error, in well-posed problems. A main counterpart for these global numerical methods, that avoid meshes, are the associated linear systems with dense and ill conditioned matrices. In these methods a sort of uncertainty principle occurs-we cannot get both accurate results and good conditioning-one of the two is lost. A specific feature of the MFS is some freedom in choosing the source points. This might lead to excellent results, but it may also lead to poor results, or even to impossible approximations. In this work we will discuss the choice of source points and propose a choice along the discrete normal direction (following [Alves CJS, Antunes PRS. The method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of 2D simply connected shapes. Comput Mater Continua 2005;2(4):251-66]), with a possible local criterion to define the distance to the boundary. We will also address some extensions that connect the asymptotic MFS to other methods by choosing the sources on a circle/sphere far from the boundary. We also present a direct connection between the approximation based on radial basis functions (RBF) and the MFS approximation in a higher dimension. This increase in dimension was somehow already present in a previous work [Alves CJS, Chen CS. A new method of fundamental solutions applied to non-homogeneous elliptic problems. Adv Comput Math 2005;23:125-42], where the frequency was used as the extra dimension. The free parameters in RBF inverse multiquadrics 2D approximation correspond in fact to the source point distance to the boundary plane in a Laplace 3D setting. Some numerical simulations are presented to illustrate theoretical issues.
机译：基本解法（MFS）可以看作是解决某些线性偏微分方程（PDE）的边值问题的最简单方法之一。它是一种无网格方法，只需少量的计算即可显示出非凡的结果。当我们需要更改域的形状时（例如在形状优化和逆问题中出现），无网格功能特别有吸引力。 MFS可以看作是Trefftz方法，其中的近似值具有验证线性PDE的优势，因此，在适当解决的问题中，我们可以将内部误差与边界误差约束在一起。这些避免网格划分的全局数值方法的主要对应方法是具有密集和病态矩阵的关联线性系统。在这些方法中，出现了一种不确定性原理-我们不能同时获得准确的结果和良好的条件-两者均会丢失。 MFS的一个特殊功能是在选择源点方面有一定的自由度。这可能会导致出色的结果，但也可能会导致结果不佳，甚至无法近似。在这项工作中，我们将讨论源点的选择，并沿离散法线方向提出选择（以下为[Alves CJS，Antunes PRS。下面的基本解法应用于二维简单连接形状的本征频率和本征模的计算。） Continua 2005; 2（4）：251-66]），并使用可能的局部准则来定义到边界的距离。我们还将通过选择远离边界的圆/球上的源来解决将渐近MFS与其他方法连接起来的一些扩展。我们还提出了基于径向基函数（RBF）的近似与更高维度的MFS近似之间的直接联系。尺寸的增加已经在以前的工作中有所体现[Alves CJS，Chen CS。一种适用于非齐次椭圆问题的基本解的新方法。 Adv Comput Math 2005; 23：125-42]，其中频率用作额外维度。实际上，RBF逆多二次2D逼近中的自由参数对应于在Laplace 3D设置中到边界平面的源点距离。提出了一些数值模拟来说明理论问题。

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