论文标题
密切评估问题的修改表示
Modified representations for the close evaluation problem
论文作者
论文摘要
当使用边界积分方程方法时，我们表示线性偏微分方程的解作为层势。众所周知，使用正交规则的层势的近似值封闭（但不在）边界上时，分辨率较差。为了应对这一挑战，我们提供了问题解决方案的修改表示。类似于用于修改拉普拉斯（Laplace）双层电位的高斯定律，我们使用了Laplace单层电位和Helmholtz层电位的修改表示，以避免近距离评估问题。某些技术是在表示公式或使用插值技术的背景下开发的。我们直接提供层势的替代修改表示（或仅当一个密度处于危险中时）。几个数值示例说明了在两个维度和三个维度上的技术效率。
When using boundary integral equation methods, we represent solutions of a linear partial differential equation as layer potentials. It is well-known that the approximation of layer potentials using quadrature rules suffer from poor resolution when evaluated closed to (but not on) the boundary. To address this challenge, we provide modified representations of the problem's solution. Similar to Gauss's law used to modify Laplace's double-layer potential, we use modified representations of Laplace's single-layer potential and Helmholtz layer potentials that avoid the close evaluation problem. Some techniques have been developed in the context of the representation formula or using interpolation techniques. We provide alternative modified representations of the layer potentials directly (or when only one density is at stake). Several numerical examples illustrate the efficiency of the technique in two and three dimensions.
