A way of using RBF as the basis for PDE's solvers is presented, its essence being constructing approximate formulas for derivatives discretizations based on RBF interpolants with local supports similar to stencils in finite difference methods. Numerical results for different types of elasticity equations showing reasonable accuracy and good h-convergence properties of the technique are presented. In particular, examples of RBF solution in the case of non-linear Karman-Fopple equations are considered. IntroductionIn numerical methods for solid mechanics, the FEM method is universally accepted technique. It allows to exploit unstructured grids thus providing considerable flexibility. However, in the recent years considerable attention was paid to the so called meshless methods which operate with nodes rather than meshes. The motivation came mainly from the following considerations:-meshless methods do not require grid generation which can be not an easy task in the three dimensional cases; -meshless methods are more appropriate than FEM or finite difference methods are in the cases of very large mesh deformation and moving discontinuities.When classifying meshless methods for solving PDEs, at least two approaches can be distinguished:(i) Methods based on the least squares type of approximations used mainly in the framework of weak formulations, approximated functions and their approximations being distinct at nodes. (ii) Methods based on the interpolation principle requiring that interpolants are equal to interpolated function at nodes.To this one can add the so called generalized finite difference method for irregularly spaced nodes [3,15,20] which uses multidimensional Taylor expansion series.There are a lot of methods of type (i) described in the literature (the relevant survey and references can be found, for example, in [4]). As to (ii), it is represented by Radial Basis Functions (RBF) techniques . Considering the latter category, the existence and accuracy of scattered data RBF interpolants are widely discussed. A large body of publications concerning the subject is presented in [13].The use of RBF for PDEs discretizations offers some nice possibilities. First, some RBF-based discretizations have potential for providing convergence rates dependent on exact solutions smoothness only rather than on degrees of underlying polynomial approximations. In certain cases, they can be exponential.Second, good RBF performance in three-dimensional cases is theoretically expected.At present, the most popular lines of attack when constructing RBF-PDE solvers seem to be collocation and boundary elements approaches [7,8,12,31].In [28,30] convergence proofs and error estimates of the collocation procedure are presented. A profound impact on the RBF-collocation technique applications is due to papers [12,14,21,27]. In the works, much attention was given to Hardy's multiquadric [11] with varying shape parameters. Dramatic increase of accuracy was found in the case of properly defined shape-parameter distributions.The main diffi...

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