Motivated by the need for the rigorous analysis of the numerical stability of variational least-squares kernel-based methods for solving second-order elliptic partial differential equations, we provide previously lacking stability inequalities. This fills a significant theoretical gap in the previous work [Comput. Math. Appl. 103 (2021) 1-11], which provided error estimates based on a conjecture on the stability. With the stability estimate now rigorously proven, we complete the theoretical foundations and compare the convergence behavior to the proven rates. Furthermore, we establish another stability inequality involving weighted-discrete norms, and provide a theoretical proof demonstrating that the exact quadrature weights are not necessary for the weighted least-squares kernel-based collocation method to converge. Our novel theoretical insights are validated by numerical examples, which showcase the relative efficiency and accuracy of these methods on data sets with large mesh ratios. The results confirm our theoretical predictions regarding the performance of variational least-squares kernel-based method, least-squares kernel-based collocation method, and our new weighted least-squares kernel-based collocation method. Most importantly, our results demonstrate that all methods converge at the same rate, validating the convergence theory of weighted least-squares in our proven theories.

翻译：受严格分析求解二阶椭圆偏微分方程的变分最小二乘核方法数值稳定性的需求驱动，我们提供了此前缺失的稳定性不等式。这填补了先前工作[Comput. Math. Appl. 103 (2021) 1-11]中的重大理论空白——该工作基于稳定性猜想给出了误差估计。现在，随着稳定性估计被严格证明，我们完善了理论基础，并将收敛行为与已证明的速率进行了比较。此外，我们建立了另一个涉及加权离散范数的稳定性不等式，并提供了理论证明，表明精确求积权重并非加权最小二乘核配置法收敛的必要条件。我们的新理论见解通过数值示例得到了验证，这些示例展示了这些方法在大网格比数据集上的相对效率和准确性。结果证实了我们对变分最小二乘核方法、最小二乘核配置法以及我们提出的新型加权最小二乘核配置法性能的理论预测。最重要的是，我们的结果表明所有方法均以相同速率收敛，从而验证了我们已证明理论中关于加权最小二乘收敛性的结论。