A Kernel-Based Least-Squares Collocation Method for Surface Diffusion

There are plenty of applications and analyses for time-independent elliptic partial differential equations in the literature hinting at the benefits of overtesting by using more collocation conditions than the number of basis functions. Overtesting not only reduces the problem size, but is also known to be necessary for stability and convergence of widely used asymmetric Kansa-type strong-form collocation methods. We consider kernel-based meshfree methods, which are methods of lines with collocation and overtesting spatially, for solving parabolic partial differential equations on surfaces without parametrization. In this paper, we extend the time-independent convergence theories for overtesting techniques to the parabolic equations on smooth and closed surfaces.