Convergence of Unsymmetric Kernel-Based Meshless Collocation Methods

This paper proves convergence of variations of the unsymmetric kernel-based collocation method introduced by Kansa in 1986. Since then, this method has been very successfully used in many applications, though it may theoretically fail in special situations, and though it had no error bound or convergence proof up to now. Thus it is necessary to add assumptions or to make modifications. Our modifications prevent numerical failure by dropping strict collocation and allow a rigorous mathematical analysis proving error bounds and convergence rates. These rates improve with the smoothness of the solution, the domain, and the kernel providing the trial spaces, but they are currently not yet optimal and deserve refinement. They are based on rates of approximation to the residuals by nonstationary meshless kernel-based trial spaces, and they are independent of the type of differential operator. The results are applicable to large classes of linear problems in strong form, provided that there is a smooth solution and the test and trial discretizations are chosen with some care. Our analysis does not require assumptions like ellipticity, and it can be extended to ill-posed problems.

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