Abstract Many local integral methods are based on an integral formulation over small and heavily overlapping stencils with local Radial Basis Functions (RBFs) interpolations. These functions have become an extremely effective tool for interpolation on scattered node sets, however the ill-conditioning of the interpolation matrix – when the RBF shape parameter tends to zero corresponding to best accuracy – is a major drawback. Several stabilizing methods have been developed to deal with this near flat RBFs in global approaches but there are not many applications to local integral methods. In this paper we present a new method called Stabilized Local Boundary Domain Integral Method (LBDIM-St) with a stable calculation of the local RBF approximation for small shape parameter that stabilizes the numerical error. We present accuracy results for some Partial Differential Equations (PDEs) such as Poisson, convection–diffusion, thermal boundary layer and an elliptic equation with variable coefficients.

中文翻译：

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椭圆偏微分方程局部边界域积分法的稳定计算

摘要 许多局部积分方法都基于具有局部径向基函数 (RBF) 插值的小型且高度重叠的模板上的积分公式。这些函数已经成为对分散节点集进行插值的极其有效的工具，但是插值矩阵的病态——当 RBF 形状参数趋于零对应于最佳精度时——是一个主要缺点。已经开发了几种稳定方法来处理全局方法中这种近乎平坦的 RBF，但是局部积分方法的应用并不多。在本文中，我们提出了一种称为稳定局部边界域积分法 (LBDIM-St) 的新方法，它可以稳定计算小形状参数的局部 RBF 近似值，从而稳定数值误差。