Convergence of one-level and multilevel unsymmetric collocation for second order elliptic boundary value problems

Abstract: Thepaperprovesconvergenceofone-levelandmultilevelunsymmetriccollocationforsecondorderelliptic boundary value problems on the bounded domains. By using Schaback's linear discretization theory,L2 errors are obtained based on the kernel-based trial spaces generated by the compactly supported radial basis functions. For the one-level unsymmetric collocation case, we obtain convergence when the testing discretization is finer than the trial discretization. The convergence rates depend on the regularity of the solution, the smoothness of the computing domain, and the approximation of scaled kernel-based spaces. The multilevel process is implemented by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Convergence of multilevel collocation is further proved based on the theoretical results of one-level unsymmetric collocation. In addition to having the same dependencies as the one-level collocation, the convergence rates of multilevel unsymmetric collocation especially depends on the increasing rules of scattered data and the selection of scaling parameters.