We propose a two dimensional (2D) adaptive nodes technique for irregular regions. The method is based on equi-distribution principal and dimension reduction. The mesh generation is carried out by first producing some adaptive nodes in a rectangle based on equi-distribution along the coordinate axes and then transforming the generated nodes to the physical domain. Since the produced mesh is applied to the meshless-type methods, the connectivity of the points is not used and only the grid points are important, though the grid lines are utilized in the adapting process. The performance of the adaptive points is examined by considering a collocation meshless method which is based on interpolation in terms of a set of radial basis functions. A generalized thin plate spline with sufficient smoothness is used as a basis function and the arc-length is employed as a monitor in the equi-distribution process. Some experimental results will be presented to illustrate the effectiveness of the proposed method.The use of radial basis functions (RBFs) for solving PDEs, first presented by Kansa, is a fully mesh free approach and falls into the domain type matheds [12,24]. This method can be easily applied to the case of higher dimensional spaces due to the nature of the RBFs. Despite a good performance of RBFs in approximating multi-variate functions, they involve ill-conditioning, especially for large scale problems. Another difficulty concerns their computational efficiency, due to the dense matrices arising from interpolation. To tackle the above difficulties some sort of localization, such as domain decomposition methods (DDM) [8] and compactly supported RBFs (CS-RBFs) [5], the most important of which was introduced by Wendland [21], have been recommended.In the DDM the domain is divided into some subdomains and the PDE is solved for each subproblem followed by assembling the global solution. As a result, the ill-conditioning is avoided and the computational efficiency is improved due to working with small size matrices. On the other hand, using the CS-RBFs results in sparse matrices, which again improve the conditioning and computational efficiency of the method.This work involves a different approach which can still be used with the above proposed methods. As was highlighted before, in using the classical RBFs, increasing the size of the problem itself affects the conditioning. Consequently, reducing the number of nodes can improve the conditioning. One way to achieve this goal is to apply a set of adaptive nodes rather than uniform ones. As is well known, the main idea in adaptive meshes is to use a minimum number of nodes while still having the desired accuracy. This is achieved by allocating more mesh points to the areas where they are required. The adaptive mesh strategies often fall into two categories: the equi-distribution principle [7] and the variational principle [23]. The most popular technique, which has been widely used in the literature, is based on the equi-distribution strategy, which is also employed in thi...

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