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：针对求解二阶椭圆型偏微分方程的变分最小二乘核方法，为满足对其数值稳定性进行严格分析的需求，本文提供了先前缺失的稳定性不等式。这填补了前期工作[Comput. Math. Appl. 103 (2021) 1-11]中基于稳定性猜想给出误差估计的重要理论空白。通过严格证明稳定性估计，我们完善了理论基础，并将收敛行为与已证实的收敛率进行比较。此外，我们建立了涉及加权离散范数的另一稳定性不等式，并给出理论证明：对于加权最小二乘核配置法而言，精确的求积权重并非其收敛的必要条件。我们通过数值算例验证了这些新颖的理论见解，这些算例展示了这些方法在大网格比数据集上的相对效率与精度。结果证实了我们对变分最小二乘核方法、最小二乘核配置法及我们提出的加权最小二乘核配置方法性能的理论预测。最重要的是，我们的结果表明所有方法均以相同速率收敛，这验证了已证明理论中加权最小二乘法的收敛理论。