A Fast Wavelet Collocation Method for Integral Equations on Polygons

We develop a fast wavelet collocation method for solving Fredholm integral equations of the second kind with weakly singular kernels on polygons, following the general setting introduced in a recent paper [11]. Specifically, we construct wavelet functions and multiscale collocation functionals having vanishing moments on polygons. Critical issues for numerical implementation of this method are considered, such as a practical matrix compression scheme, numerical integration of weakly singular integrals, error controls of numerical quadratures and numerical solutions of the resulting compressed linear systems. The estimate of the computational complexity ensures the proposed method is a fast algorithm. Numerical experiments are presented to demonstrate the proposed methods and confirm the theoretical estimates.

[1]  Weifu Fang,et al.  Multi-Level Iteration Methods for Solving Integral Equations of the Second Kind , 2002 .

[2]  Andreas Rathsfeld,et al.  A Wavelet Algorithm for the Solution of a Singular Integral Equation over a Smooth Two-Dimensional Manifold , 1998 .

[3]  Wolfgang Dahmen,et al.  Wavelet approximation methods for pseudodifferential equations: I Stability and convergence , 1994 .

[4]  B. Alpert.  A class of bases in L 2 for the sparse representations of integral operators , 1993 .

[5]  Christoph Schwab,et al.  Wavelet approximations for first kind boundary integral equations on polygons , 1996 .

[6]  Yi Wang,et al.  An Implementation of Fast Wavelet Galerkin Methods for Integral Equations of the Second Kind , 2004, J. Sci. Comput..

[7]  Hideaki Kaneko,et al.  Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind , 1994 .

[8]  Christoph Schwab.  Variable order composite quadrature of singular and nearly singular integrals , 2005, Computing.

[9]  Kendall E. Atkinson,et al.  The Collocation Method for Solving the Radiosity Equation for Unoccluded Surfaces , 1998 .

[10]  Kendall E. Atkinson,et al.  A Survey of Boundary Integral Equation Methods for the Numerical Solution of Laplace’s Equation in Three Dimensions , 1990 .

[11]  Ian H. Sloan,et al.  Piecewise Continuous Collocation for Integral Equations , 1983 .

[12]  陈仲英,et al.  MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS , 2005 .

[13]  Kendall E. Atkinson,et al.  Piecewise Polynomial Collocation for Boundary Integral Equations , 1995, SIAM J. Sci. Comput..

[14]  Wolfgang Dahmen,et al.  A multiscale method for the double layer potential equation on a polyhedron. , 2005 .

[15]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[16]  Yuesheng Xu,et al.  The Petrov–Galerkin method for second kind integral equations II: multiwavelet schemes , 1997, Adv. Comput. Math..

[17]  C. Micchelli,et al.  Using the Matrix Refinement Equation for the Construction of Wavelets on Invariant Sets , 1994 .

[18]  Charles A. Micchelli,et al.  A Construction of Refinable Sets for Interpolating Wavelets , 1998 .