Abstract
In this paper is discussed solving an elliptic equation and a boundary integral equation of the second kind by representation of compactly supported wavelets. By using wavelet bases and the Galerkin method for these equations, we obtain a stiff sparse matrix that can be ill-conditioned. Therefore, we have to introduce an operator which maps every sparse matrix to a circulant sparse matrix. This class of circulant matrices is a class of preconditioners in a Banach space. Based on having some properties in the spectral theory for this class of matrices, we conclude that the circulant matrices are a good class of preconditioners for solving these equations. We called them circulant wavelet preconditioners (CWP). Therefore, a class of algorithms is introduced for rapid numerical application.
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Rostami Varnos Fadrani, D. Circulant Wavelet Preconditioners for Solving Elliptic Differential Equations and Boundary Integral Equations. Journal of Computational Analysis and Applications 5, 255–271 (2003). https://doi.org/10.1023/A:1022885129626
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DOI: https://doi.org/10.1023/A:1022885129626