标题： 惠特尼方法的基本解与卢辛小波 显示英文标题

标题： The Whitney method of fundamental solutions with Lusin wavelets

摘要： 我们为一种基本解方法（MFS）的变体建立了理论基础，其中源点 $\{q_j\}_{j=1}^\infty$ 以惠特尼方式向域内聚集，这意味着它们之间的分离距离与到域的距离成正比。 我们证明了归一化的卢辛小波 $\psi_j(w) = b_j(w-q_j)^{-2}$ 构成了域上全纯函数的 $L_2$-迹的哈代子空间的一个广义基，即框架。 因此，我们的方法——其中使用 $\psi_j$ 作为MFS中的基函数——能够对拉普拉斯边值问题的解实现数值稳定的逼近，即使这些解在边界上没有解析延拓。 尽管源点向域内聚集，我们的计算在整个边界范围内至少可以达到12位的精度，包括当解没有解析延拓或边界有角点的情况。 显示英文摘要

摘要： We establish the theoretical foundation for a variant of the method of fundamental solutions (MFS), where the source points $\{q_j\}_{j=1}^\infty$ accumulate towards the domain in a Whitney fashion, meaning that their separation is proportional to the distance to the domain. We prove that the normalized Lusin wavelets $\psi_j(w) = b_j(w-q_j)^{-2}$ constitute a generalized basis, known as a frame, for the Hardy subspace of $L_2$-traces of holomorphic functions on the domain. Consequently, our method, where $\psi_j$ are used as basis functions in the MFS, enables a numerically stable approximation of solutions to Laplace boundary value problems, even when the solutions lack analytic continuation across the boundary. Despite the source points accumulating towards the domain, our computations achieve at least 12 digits of accuracy uniformly up to the boundary, including cases when the solution lacks analytic continuation or when the boundary has corners.