Local meshless methods using RBFs augmented with monomials have become increasingly popular, due to the fact that they can be used to solve PDEs on scattered node sets in a dimension-independent way, with the ability to easily control the order of the method, but at a greater cost to execution time. We analyze this ability on a Poisson problem with mixed boundary conditions in 1D, 2D and 3D, and reproduce theoretical convergence orders practically, also in a dimension-independent manner, as demonstrated with a solution of Poisson’s equation in an irregular 4D domain. The results are further combined with theoretical complexity analyses and with conforming execution time measurements, into a study of accuracy versus execution time trade-off for each dimension. Optimal regimes of order for given target accuracy ranges are extracted and presented, along with guidelines for generalization.

使用RBF加上单项式的局部无网格方法已变得越来越普遍，这是因为它们可以以独立于维的方式用于解决散乱节点集上的PDE，并且能够轻松控制方法的顺序，但是花费更多的执行时间。我们在1D，2D和3D混合边界条件下的Poisson问题上分析了这种能力，并且还以不依赖维度的方式实际再现了理论收敛阶，如在不规则4D域中的Poisson方程的解所示。将结果与理论复杂度分析和一致的执行时间测量结果进一步组合，以研究每个维度的精度与执行时间之间的权衡。提取并给出针对给定目标精度范围的最优顺序方案，