论文标题
倒数近似和平面PDE求解器
Reciprocal-log approximation and planar PDE solvers
论文作者
论文摘要
本文涉及近似理论和部分微分方程（PDE）的数值解。首先，我们介绍{\ em倒数 - log}的概念，或通过$ g（z）= \ sum_k c_k c_k c_k /（z-log log（z- z_k）$的函数，分支点奇点的分析函数的分析函数的{\ em log-lightning近似} $ g（z_k \} $） 表面。我们证明，最佳倒数近似值的错误相对于$ n $呈指数下降，并且指数或近乎指数的收敛性（即，以$ o（\ exp（-c n / \ log n）$的速率）也以近距离的近似为基础，由lineare的近似近似近似构建。然后，我们将这些结果应用于带有角奇异点的二维域中拉普拉斯和相关PDE的数值解和相关PDE的数值解。与基于理性函数的原始闪电方法的根指数收敛相反，收敛是几乎指数的。
This article is about both approximation theory and the numerical solution of partial differential equations (PDEs). First we introduce the notion of {\em reciprocal-log} or {\em log-lightning approximation} of analytic functions with branch point singularities at points $\{z_k\}$ by functions of the form $g(z) = \sum_k c_k /(\log(z-z_k) - s_k)$, which have $N$ poles potentially distributed along a Riemann surface. We prove that the errors of best reciprocal-log approximations decrease exponentially with respect to $N$ and that exponential or near-exponential convergence (i.e., at a rate $O(\exp(-C N / \log N))$) also holds for near-best approximations with preassigned singularities constructed by linear least-squares fitting on the boundary. We then apply these results to derive a "log-lightning method" for numerical solution of Laplace and related PDEs in two-dimensional domains with corner singularities. The convergence is near-exponential, in contrast to the root-exponential convergence for the original lightning methods based on rational functions.
