Abstract We propose to solve elliptic interface problems by a meshless finite difference method, where the second order elliptic operator and jump conditions are discretized with the help of the QR decomposition of an appropriately rescaled multivariate Vandermonde matrix with partial pivoting. A prescribed consistency order is achieved on irregular nodes with small influence sets, which allows to place the nodes directly on the unfitted interface and leads to sparse system matrices with the density of nonzero entries comparable to the density of the system matrices arising from the mesh-based finite difference or finite element methods. Numerical experiments on a number of standard test problems with known solutions demonstrate convergence orders up to O ( h 6 ) for both the approximate solution and its gradient, and a robust performance of the method in the case when the interface is known inaccurately.

中文翻译：

---

基于旋转二维分解的椭圆界面问题的无网格有限差分法

摘要 我们建议通过无网格有限差分方法解决椭圆界面问题，其中二阶椭圆算子和跳跃条件在具有部分旋转的适当重新缩放的多元 Vandermonde 矩阵的 QR 分解的帮助下被离散化。在具有小影响集的不规则节点上实现了规定的一致性顺序，这允许将节点直接放置在未拟合的界面上，并导致非零条目密度与网格产生的系统矩阵密度相当的稀疏系统矩阵 -基于有限差分或有限元方法。对许多具有已知解的标准测试问题的数值实验证明了近似解及其梯度的收敛阶数高达 O ( h 6 )，