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.

翻译：本文在有限域上证明了二阶椭圆边值问题中单层与多层非对称配置法的收敛性。通过采用Schaback线性离散化理论，基于紧支径向基函数生成的核函数试验空间，获得了L²误差估计。对于单层非对称配置情形，当检验离散化精度高于试验离散化时，我们推导出收敛性。收敛速率取决于解的 regularity、计算域的光滑性以及缩放核函数空间的逼近能力。多层过程通过采用逐步细化的散乱数据集和具有变支撑半径的缩放紧支径向基函数实现。基于单层非对称配置的理论结果，进一步证明了多层配置法的收敛性。与单层配置相比，多层非对称配置的收敛速率不仅依赖相同因素，还特别取决于散乱数据的递增规则与缩放参数的选取。