RBF-FD方法中的超收敛结果
A superconvergence result in the RBF-FD method
AI总结：
该研究揭示了RBF-FD方法中通过单项式补充可实现的超收敛现象，展示了在特定条件下收敛阶数的提升。
摘要：
径向基函数生成有限差分（RBF-FD）是一种无网格方法，可用于数值求解偏微分方程。求解过程包括两个步骤。首先，在给定的散节点上对微分算子进行离散化，随后组装并求解一个全局稀疏矩阵以获得近似解。聚焦于多项式谐波样条作为所选径向基函数（RBFs），适当补充单项式，已知微分算子近似的截断误差由单项式的次数决定。直观上，人们可能会认为解误差的收敛阶数相同。我们提出了一个超收敛结果，表明情况并非如此——对于某些补充次数，收敛阶数高于预期。
英文摘要：
Radial Basis Function-generated Finite Differences (RBF-FD) is a meshless method that can be used to numerically solve partial differential equations. The solution procedure consists of two steps. First, the differential operator is discretised on given scattered nodes and afterwards, a global sparse matrix is assembled and inverted to obtain an approximate solution. Focusing on Polyharmonic Splines as our Radial Basis Functions (RBFs) of choice, appropriately augmented with monomials, it is well known that the truncation error of the differential operator approximation is determined by the degree of monomial augmentation. Naively, one might think that the solution error will have the same order of convergence. We present a superconvergence result that shows otherwise - for some augmentation degrees, order of convergence is higher than expected.
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