On the Asymptotics of Least-squares Kernel Collocation Methods

Kernel collocation methods are meshfree and enjoy advantages in that they do not require construction of mesh grids. Since its inception, meshless methods have been welcomed by many researchers and successfully applied to solve a vast amount of applications and simulations in irregular domains. The aim of this research project is to develop some continuous least-squares kernel collocation methods and numerical techniques capable of yielding efficient solvers for elliptic partial differential equations with low regularity.

In discrete least-squares kernel collocation methods, a fixed weight depending on some denseness measures of data points is imposed on all boundary collocation conditions. We proved that this discrete approach is convergent for a class of kernels that reproduce Sobolev spaces [SIAM Numer. Anal., 56(1):614-633. 2018]. However, using a weight that ignores local geometries of data points is not numerically sound when data points are highly nonuniform; but, this is exactly what we expect from any adaptive method for solving equations with non-smooth coefficients, forcing functions, and/or boundary conditions. We propose to look into the asymptotic of the discrete approach as refined collocation points and consider continuous least-squares formulations. The first task of this project is to numerically identify some collocation-pointwise weights that allow kernel collocation methods to solve problems with discontinuous coefficients.

The next challenging task is to come up with a rigid convergence analysis for the continuous least-squares kernel collocation methods. We note that regularity estimates for elliptic problems do not naturally yield upper bounds in L2 norms and leave a gap between the continuous least-squares formulations. We conjecture an inverse inequality for trial functions. Convergence proof will be built on top of this result.

In the last part of this project, we will develop some boundary-type kernel collocation methods that will be an alternative to the method of fundamental solutions but without the restriction on the differential operators. We propose to extend the integration domain for the interior residual to some larger domains that allow either analytic or easy numerical integrations. The former is computationally efficient, but restricted by our knowledge of analytical solutions to some integrals. The latter is more general, but comes with an overhead of numerical quadrature. All developed algorithms will adopt a fast direct matrix factorization algorithm and be subjected to thorough numerical experiments for understanding their limitations.