  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.
