A compression technique for the boundary integral equation reduced from Helmholtz equation

References

• Benamou , J. D. , Lafitte , O. , Solliec , I. and Sentis , R. 2004 . A geometric optics method for high-frequency electromagnetic fields computations near fold caustics: II. The energy . J. Comput. Appl. Math. , 167 ( 1 ) : 91 – 134 . (doi:10.1016/j.cam.2003.09.046)
   Web of Science ®Google Scholar
• Beylkin , G. , Coifman , R. and Rokhlin , V. 1991 . Fast wavelet transforms and numerical algorithms I . Comm. Pure Appl. Math. , 44 : 141 – 183 . (doi:10.1002/cpa.3160440202)
   Web of Science ®Google Scholar
• Chen , J. 2012 . Fast multilevel augmentation method for nonlinear integral equations . Int. J. Comput. Math. , 89 ( 1 ) : 80 – 89 . (doi:10.1080/00207160.2011.627436)
   Web of Science ®Google Scholar
• Chen , Z. and Liu , X. 2005 . An adaptive perfectly matched layer technique for time-harmonic scattering problems . SIAM J. Numer. Anal. , 43 ( 2 ) : 645 – 671 . (doi:10.1137/040610337)
   Web of Science ®Google Scholar
• Chen , Z. , Micchelli , C. A. and Xu , Y. 1997 . The Petrov-Galerkin methods for second kind integral equations II: Multiwavelet scheme . Adv. Comput. Math. , 7 : 199 – 233 . (doi:10.1023/A:1018994802659)
   Web of Science ®Google Scholar
• Chen , Z. , Micchelli , C. A. and Xu , Y. 2002 . Fast collocation methods for second kind integral equations . SIAM J. Numer. Anal. , 40 : 344 – 375 . (doi:10.1137/S0036142901389372)
   Web of Science ®Google Scholar
• Chen , Z. , Cheng , S. , Nelakanti , G. and Yang , H. 2010 . A fast multiscale Galerkin method for the first kind ill-posed integral equations via Tikhonov regularization . Int. J. Comput. Math. , 87 ( 3 ) : 565 – 582 . (doi:10.1080/00207160802155302)
   Web of Science ®Google Scholar
• Colton , D. and Kress , R. 1998 . Inverse Acoustic and Electromagnetic Scattering Theory , Berlin , , Heidelberg : Springer-Verlag .
   Google Scholar
• Dahmen , W. , Harbrecht , H. and Schneider , R. 2006 . Compression techniques for boundary integral equations-asymptotically optimal complexity estimates . SIAM J. Numer. Anal. , 43 ( 6 ) : 2251 – 2271 . (doi:10.1137/S0036142903428852)
   Web of Science ®Google Scholar
• Huybrechs , D. , Simoens , J. and Vandewalle , S. 2004 . A note on wave number dependence of wavelet matrix compression for integral equations with oscillatory kernel . J. Comput. Appl. Math. , 172 : 233 – 246 . (doi:10.1016/j.cam.2004.02.006)
   Web of Science ®Google Scholar
• Kreigsmann , G. A. and Morawetz , C. S. 1980 . Solving the Helmholtz equation for exterior problems with variable index of refraction: I . SIAM J. Sci. Stat. Comput. , 1 : 371 – 385 . (doi:10.1137/0901026)
   Web of Science ®Google Scholar
• Kumar , R. B.V. and Mehra , M. 2006 . A three-step wavelet Galerkin method for parabolic and hyperbolic partial differential equations . Int. J. Comput. Math. , 83 : 143 – 157 . (doi:10.1080/00207160500112985)
   Web of Science ®Google Scholar
• Kussmaul , R. 1969 . Ein numerisches Verfahren zur Lösung des Neumannschen Aussenraumproblems für die Helmholtzsche Schwingungsgleichung . Computing , 4 : 246 – 273 . (doi:10.1007/BF02234773)
   Web of Science ®Google Scholar
• Langdon , S. and Chandler-Wilde , S. N. 2006 . A wavenumber independent boundary element method for an acoustic scattering problem . SIAM J. Numer. Anal. , 43 ( 6 ) : 2450 – 2477 . (doi:10.1137/S0036142903431936)
   Web of Science ®Google Scholar
• Li , S. H. 2011 . A fast trigonometric wavelets method for the biharmonic equation . Appl. Anal. , 90 ( 8 ) : 1203 – 1214 . (doi:10.1080/00036810903277119)
   Web of Science ®Google Scholar
• Li , S. H. and Lin , W. 2005 . A numerical method based on the generalized Shannon–Whittaker representation theorem for solving Helmholtz equation . Complex Var. Theory Appl. , 50 ( 7 ) : 893 – 911 .
   Google Scholar
• Li , S. H. and Sun , M. B. 2009 . A fast numerical method for a natural boundary integral equation for the Helmholtz equation . J. Comput. Appl. Math. , 230 : 341 – 350 . (doi:10.1016/j.cam.2008.12.002)
   Web of Science ®Google Scholar
• Li , S. H. , Lin , W. and Sun , M. B. 2008 . A fast collocation method for acoustic scattering in shallow oceans . J. Math. Anal. Appl. , 347 : 607 – 620 . (doi:10.1016/j.jmaa.2008.06.023)
   Web of Science ®Google Scholar
• Martensen , E. 1963 . Über eine Methode zum räumlichen Neumannschen Problem mit einer Anwendung für torusartige Berandungen . Acta Math. , 109 : 75 – 135 . (doi:10.1007/BF02391810)
   Web of Science ®Google Scholar
• Yang , X. , Xu , D. and Zhang , H. 2011 . Quasi-wavelet based numerical method for fourth-order partial integro-differential equations with a weakly singular kernel . Int. J. Comput. Math. , 88 ( 15 ) : 3236 – 3254 . (doi:10.1080/00207160.2011.587003)
   Web of Science ®Google Scholar
• Zheng , X. , Yang , X. , Su , H. and Qiu , L. 2011 . Mixed discontinuous Legendre wavelet Galerkin method for solving elliptic partial differential equations . Int. J. Comput. Math. , 88 ( 17 ) : 3626 – 3645 . (doi:10.1080/00207160.2011.610449)
   Web of Science ®Google Scholar