Weighted least-squares collocation methods for elliptic PDEs with mixed boundary conditions

In this paper, we apply kernel-based collocation methods to elliptic problems with mixed boundary conditions. We propose some weighted least-squares formulations with different weights for the Dirichlet and Neumann boundary collocation terms. Besides fill distance of discrete sets, our weights also depend on other three factors: the proportion of measures of the Dirichlet and Neumann boundaries, dimensionless volume ratios of the boundary and domain, and kernel smoothness. We determine the dependencies of these terms in weights by different numerical tests. Our least-squares formulations can be proved convergent in H²(Ω). Numerical experiments for two dimensional examples show that we can obtain convergent solutions for kernel smoothness m ∈ {3, 4, 5} in the irregular domains, circle domain, and rectangle thin domain. We also apply our formulations to three dimensional cases and get desired convergent results for m ∈ {4, 5, 6, 7} in cubic, sphere and torus domain under different boundary conditions.