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The strong-form asymmetric kernel-based collocation method, commonly referredto as the Kansa method, is easy to implement and hence is widely used forsolving engineering problems and partial differential equations despite thelack of theoretical support. The simple least-squares (LS) formulation, on theother hand, makes the study of its solvability and convergence rathernontrivial. In this paper, we focus on general second order linear ellipticdifferential equations in $\Omega \subset R^d$ under Dirichlet boundaryconditions. With kernels that reproduce $H^m(\Omega)$ and some smoothnessassumptions on the solution, we provide denseness conditions for a constrainedleast-squares method and a class of weighted least-squares algorithms to beconvergent. Theoretically, we identify some $H^2(\Omega)$ convergent LSformulations that have an optimal error behavior like $h^{m-2}$. We alsodemonstrate the effects of various collocation settings on the respectiveconvergence rates, as well as how these formulations perform with high orderkernels and when coupled with the stable evaluation technique for the Gaussiankernel.

$H^2$--Convergence of least-squares kernel collocation methods