Wavelet Based Adaptive RBF Method for Nearly Singular Poisson-Type Problems on Irregular Domains
Nicolas Ali Libre; Arezoo Emdadi; Edward J. Kansa;
Mohammad Shekarchi and Mohammad Rahimian

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 50, No. 2, pp. 161-190, 2009
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Keywords Adaptive node refinement, meshless, RBF collocation, Wavelet decomposition, Poisson-type equation, irregular domain, nearly singular PDEs, Multiquadrics
Abstract We present a wavelet based adaptive scheme and investigate the efficiency of this scheme for solving nearly singular potential PDEs over irregularly shaped domains. For a problem defined overWd, the boundary of an irregularly shaped domain,G, is defined as a boundary curve that is a product of a Heaviside function along the normal direction and a piecewise continuous tangential curve. The link between the original wavelet based adaptive method presented in Libre, Emdadi, Kansa, Shekarchi, and Rahimian (2008, 2009) or LEKSR method and the generalized one is given through the use of simple Heaviside masking procedure. In addition level dependent thresholding were introduced to improve the efficiency and convergence rate of the solution. We will show how the generalized wavelet based adaptive method can be applied for detecting nearly singularities in Poisson type PDEs over irregular domains. The numerical examples have illustrated that the proposed method is powerful to analyze the Poisson type PDEs with rapid changes in gradients and nearly singularities.
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