标题： 威特尼基本解方法与卢辛小波 显示英文标题

标题： The Whitney method of fundamental solutions with Lusin wavelets

摘要： 我们建立了方法的基本解 (MFS) 的一种变体的理论基础，其中源点 $\{q_j\}_{j=1}^\infty$ 以 Whitney 方式聚集到区域内部，这意味着它们之间的距离与到区域的距离成比例。 我们证明了归一化的 Lusin 小波 $\psi_j(w) = b_j(w-q_j)^{-2}$ 构成了该区域上全纯函数的 $L_2$-迹的 Hardy 子空间的一个广义基，即一个框架。 因此，我们的方法中，使用 $\psi_j$ 作为 MFS 中的基函数，能够在解无法在边界上解析延拓的情况下，对 Laplace 边界值问题的解进行数值稳定的近似。 尽管源点聚集到区域附近，我们的计算在接近边界时仍能实现至少 12 位精度的均匀计算，包括当解无法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.