  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.
