Abstract
We present a numerical implementation of the fast Galerkin method for Fredholm integral equations of the second kind using the piecewise polynomial wavelets. We focus on addressing critical issues for the numerical implementation of such a method. They include a choice of practical truncation strategy, numerical integration of weakly singular integrals and the error control of the numerical quadrature. We also implement a multiscale iteration method for solving the resulting compressed linear system. Numerical examples are given to demonstrate the proposed ideas and methods.
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REFERENCES
Alpert, B. (1993). A class of bases in L2 for the sparse representation of integral operator. SIAM J. Math. Anal. 24, 246–262.
Atkinson, K. E. (1997). The Numerical Solution of Integral Equations of Second Kind, Cambridge University Press, Cambridge.
Beylkin, G., Coifman, R., and Rokhlin, V. (1991). Fast wavelet transforms and numerical algorithms I. Comm. Pure Appl. Math. 44, 141–183.
Chen, Z., Micchelli, C. A., and Xu, Y. (2001). A multilevel method for solving operator equations. J. Math. Anal. Appl. 262, 688–699.
Dahmen, W., Kleemann, B., Prössdorf, S., and Schneider, R. (1994). A multiscale method for the double layer potential equation on a polyhedron. In Dikshit, H. P., and Micchelli, C. A. (eds.), Advances in Computational Mathematics: New Delhi, India, World Scientific, Singapore.
Dahmen, W., Prössdorf, S., and Schneider, R. (1994). Wavelet approximation methods for psuedodifferential equations I: Stability and convergence. Math. Z. 215, 583–620.
Dahmen, W., Prössdorf, S., and Schneider, R. (1993). Wavelet approximation methods for psuedodifferential equations II: Matrix compression and fast solutions. Adv. Comput. Math. 1, 259–335.
Fang, W., Ma, F., and Xu, Y. (2002). Multilevel iteration methods for solving integral equations of the second kind. J. Integral Equations Appl. 14, 355–376.
Kaneko, H., and Xu, Y. (1994). Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind. Math. Comp. 62, 739–753.
Kress, R. (1999). Linear Integral Equations, 2nd ed., Springer-Verlag, New York.
Kress, R. (1990). A Nyström method for boundary integral equations in domains with corners. Numer. Math. 58, 145–161.
Micchelli, C. A., and Xu, Y. (1994). Using the matrix refinement equation for the construction of wavelets on invariant sets. Appl. Comput. Harmon. Anal. 1, 391–401.
Micchelli, C. A., Xu, Y., and Zhao, Y. (1997). Wavelet Galerkin methods for second-kind integral equations. J. Comp. Appl. Math. 86, 251–270.
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Fang, W., Wang, Y. & Xu, Y. An Implementation of Fast Wavelet Galerkin Methods for Integral Equations of the Second Kind. Journal of Scientific Computing 20, 277–302 (2004). https://doi.org/10.1023/B:JOMP.0000008723.85496.ce
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DOI: https://doi.org/10.1023/B:JOMP.0000008723.85496.ce