High-order close evaluation of Laplace layer potentials: A differential geometric approach
High-order close evaluation of Laplace layer potentials: A differential geometric approach
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DOI：
10.1137/21m1423051
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发表时间：
2021-05
期刊：
影响因子：
--
通讯作者：
Hai-Ping Zhu;S. Veerapaneni
中科院分区：
文献类型：
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作者：
Hai-Ping Zhu;S. Veerapaneni
本文提出了一种新的方法来解决三维封闭评估问题，经常遇到的线性椭圆型偏微分方程，通过潜在的理论。其目的是评估层电位接近的边界上，他们被定义。这里介绍的方法将这些近奇异积分的补丁边界上的一组非奇异线积分的补丁边界上使用流形上的斯托克斯定理。基于调和多项式的函数逼近方案的目的是表达的形式，是适合应用斯托克斯定理的被积函数。只要数据-边界和密度函数-以高阶格式给出，双层势及其导数可以使用该方案在边界上和边界外以高阶精度进行计算。特别是，我们提出的数值计算结果表明，七阶收敛的光滑，扭曲的环面的例子，实现10位数的精度，在任意接近边界的目标，在评估双层潜力。
This paper presents a new approach for solving the close evaluation problem in three dimensions, commonly encountered while solving linear elliptic partial differential equations via potential theory. The goal is to evaluate layer potentials close to the boundary over which they are defined. The approach introduced here converts these nearly-singular integrals on a patch of the boundary to a set of non-singular line integrals on the patch boundary using the Stokes theorem on manifolds. A function approximation scheme based on harmonic polynomials is designed to express the integrand in a form that is suitable for applying the Stokes theorem. As long as the data -- the boundary and the density function -- is given in a high-order format, the double-layer potential and its derivatives can be evaluated with high-order accuracy using this scheme both on and off the boundary. In particular, we present numerical results demonstrating seventh-order convergence on a smooth, warped torus example achieving 10-digit accuracy in evaluating double layer potential at targets that are arbitrarily close to the boundary.