New RBF Collocation Methods and Kernel RBF with Applications
A few novel radial basis function (RBF) discretization schemes for partial differential equations are developed in this study. For boundary-type methods, we derive the indirect and direct symmetric boundary knot methods. Based on the multiple reciprocity
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Introduction
Many existing meshfree methods require using the moving least square (MLS). Exceptionally, the numerical schemes based on the radial basis function (RBF) do not need the MLS at all and are inherently meshfree. For some new advances on the RBF see Buhmann's excellent survey [1). The RBF has physical backgrounds of field and potential theory [2) and is justified mathematically by integral equation theory [3). Among RBF numerical schemes, famous are the Kansa method [4], Hermite symmetric RBF collocation method [5,6) and the method of fundamental solution (MFS) [7). The Kansa's method is the very first domain-type RBF collocation scheme with easy-to-use merit, but the method lacks symmetric interpolation matrix due to boundary collocation. The Hermite RBF collocation method kills the unsymmetrical drawback. Like the Kansa's method, however, the method suffers relatively lower accuracy in boundary-adjacent region. The MFS, also known as the regular BEM, is a simple and efficient boundary-type RBF scheme, but the controversial artificial boundary outside physical domain hinders its practical applications. The boundary knot method, recently introduced by the present author [2,3], surpasses the MFS in that it employs the nonsingular general solution instead of the singular fundamental solution and thus no longer requires the arbitrary fictitious boundary. Albeit better than the MKM, the BKM loses symmetric merit whenever the presence of mixed boundary conditions. It is worth pointing out that all these RBF schemes are indirect and global. The indirect methods mean that the expansion coefficients rather than physical variables are used as the basic variable, while the global interpolation causes the ill-conditioning interpolation matrix and susceptible to Gibbs phenomenon amid weak continuity of physical solution. M. Griebel et al. (eds.), Meshfree Methods for Partial Differential Equations © Springer-Verlag Berlin Heidelberg 2003
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W. Chen
The purpose of this paper is to introduce a few new RBF discretization schemes of boundary and domain types to overcome the aforementioned shortcomings. On the other hand, the proper RBF is also an essential issue leading to an efficient and stable solution. In general, there is not an operational approach to create efficient RBF available now. Based on the underlying relationship between the RBF and the Green integral [3], we summarize five approaches constructing kernel RBF [8], which also cover all existing popular RBFs. The rest of this paper is structured as follows. In section 2, we establish the symmetric BKM and the direct BKM, and then, the symmetric boundary particle method (BPM) is developed by using the multiple reciprocity principle [9]. Unlike the BKM, the BPM does not require the interior nodes for inhomogeneous problems. By using Green integral, section 3 presents the symmetric modified Kansa's method (MKM) to significantly improve the solution accuracy at nodes neighboring boundary. Furthermore, we briefly describe the spline version of the MKM cal
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