Abstract
A general framework for proving error bounds and convergence of a large class of unsymmetric meshless numerical methods for solving well-posed linear operator equations is presented. The results provide optimal convergence rates, if the test and trial spaces satisfy a stability condition. Operators need not be elliptic, and the problems can be posed in weak or strong form without changing the theory. Non-stationary kernel-based trial and test spaces are shown to fit into the framework, disregarding the operator equation. As a special case, unsymmetric meshless kernel-based methods solving weakly posed problems with distributional data are treated in some detail. This provides a foundation of certain variations of the “Meshless Local Petrov-Galerkin” technique of S.N. Atluri and collaborators.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.