A Fast Adaptive Wavelet scheme in RBF Collocation for nearly singular potential PDEs
Nicolas Ali Libre; Arezoo Emdadi; Edward J. Kansa; Mohammad Shekarchi; Mohammad Rahimian

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 38, No. 3, pp. 263-284, 2008
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Keywords RBF collocation, Wavelet decomposition, Multiresolution analysis, Adaptive distribution, Potential problem, Multiquadrics
Abstract We present a wavelet based adaptive scheme and investigate the efficiency of this scheme for solving nearly singular potential PDEs. Multiresolution wavelet analysis (MRWA) provides a firm mathematical foundation by projecting the solution of PDE onto a nested sequence of approximation spaces. The wavelet coefficients then were used as an estimation of the sensible regions for node adaptation. The proposed adaptation scheme requires negligible calculation time due to the existence of the fast Discrete Wavelet Transform (DWT). Certain aspects of the proposed adaptive scheme are discussed through numerical examples. It has been shown that the proposed adaptive scheme can detect the singularities both in the domain and near the boundaries. Moreover, the proposed adaptive scheme can be utilized for capturing the regions with high gradient both in the solution and its spatial derivatives. Due to the simplicity of the proposed method, it can be efficiently applied to large scale nearly singular engineering problems.
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