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Section B

The superconvergence of composite trapezoidal rule for Hadamard finite-part integral on a circle and its application

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Pages 855-876 | Received 04 Dec 2007, Published online: 02 Oct 2008

References

  • Bao , G. and Sun , W. 2005 . A fast algorithm for the electromagnetics scattering from a cavity . SIAM J. Sci. Comput. , 27 : 553 – 574 .
  • Bialecki , B. and Fairweather , G. 2001 . Orthogonal spline collocation methods for partial differential equations . J. Comput. Appl. Math. , 128 : 55 – 82 .
  • Brunner , H. and Yan , N. 1996 . On global superconvergence of iterated collocation solutions to linear second-kind Volterra integral equations . J. Comput. Appl. Math. , 67 : 185 – 189 .
  • Chen , Y. Z. 2004 . Hypersingular integral equation method for three-dimensional crack problem in shear mode . Commun. Numer. Methods Eng. , 20 : 441 – 454 .
  • Chen , J. T. and Hong , H. K. 1999 . Review of dual boundary element methods with emphasis on hypersingular integral and divergent series . Appl. Mech. Rev. , 52 : 17 – 33 .
  • Choi , U. J. , Kim , S. W. and Yun , B. I. 2004 . Improvement of the asymptotic behaviour of the Euler–Maclaurin formula for Cauchy principal value and Hadamard finite-part integrals . Int. J. Numer. Methods Eng. , 61 : 496 – 513 .
  • Criscuolo , G. 1997 . A new algorithm for Cauchy principal value and Hadamard finite-part integrals . J. Comput. Appl. Math. , 78 : 255 – 275 .
  • Du , Q. K. 2001 . Evaluations of certain hypersingular integrals on interval . Int. J. Numer. Methods Eng. , 51 : 1195 – 1210 .
  • Du , Q. K. and Yu , D. H. 2002 . A domain decomposition method based on natural boundary reduction for nonlinear time-dependent exterior wave problems . Computing , 68 : 111 – 129 .
  • Du , Q. K. and Yu , D. H. 2003 . Dirichlet–Neumann alternating algorithm based on the natural boundary reduction for the time-dependent problems over an unbounded domain . Appl. Numer. Math. , 44 : 471 – 486 .
  • Elliott , D. and Venturino , E. 1997 . Sigmoidal transformations and the Euler–Maclaurin expansion for evaluating certain Hadamard finite-part integrals . Numer. Math. , 77 : 453 – 465 .
  • Feng , K. 1983 . “ Finite element method and natural boundary reduction ” . In Proceeding of the International Congress of Mathematicians 1439 – 1453 . Warszawa
  • Han , H. and Wu , X. 1985 . Approximation of infinite boundary condition and its application to finite element methods . J. Comput. Math. , 3 : 179 – 192 .
  • Hasegawa , T. 2004 . Uniform approximations to finite Hilbert transform and its derivative . J. Comput. Appl. Math. , 163 : 127 – 138 .
  • Hui , C. Y. and Shia , D. 1999 . Evaluations of hypersingular integrals using Gaussian quadrature . Int. J. Numer. Methods Eng. , 44 : 205 – 214 .
  • Ioakimidis , N. I. 1985 . On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives . Math. Comput. , 44 : 191 – 198 .
  • Kim , P. and Jin , U. C. 2003 . Two trigonometric quadrature formulae for evaluating hypersingular integrals . Int. J. Numer. Methods Eng. , 56 : 469 – 486 .
  • Koyama , D. 2007 . Error estimates of the DtN finite element method for the exterior Helmholtz problem . J. Comput. Appl. Math. , 200 : 21 – 31 .
  • Kress , R. 1995 . On the numerical solution of a hypersingular integral equation in scattering theory . J. Comput. Appl. Math. , 61 : 345 – 360 .
  • Li , R. X. 1999 . On the coupling of BEM and FEM for exterior problems for the Helmholtz equation . Math. Comput. , 68 : 945 – 953 .
  • Linz , P. 1985 . On the approximate computation of certain strongly singular integrals . Computing , 35 : 345 – 353 .
  • Monegato , G. 1994 . Numerical evaluation of hypersingular integrals . J. Comput. Appl. Math. , 50 : 9 – 31 .
  • Nedelec , J. C. 1982 . Integral equations with non-integrable kernels . Integral Eq. Oper. Theory , 5 : 562 – 572 .
  • Nicholls , D. P. and Nigam , N. 2006 . Error analysis of an enhanced DtN-FE method for exterior scattering problems . Numer. Math. , 105 : 267 – 298 .
  • Paget , D. F. 1981 . The numerical evaluation of Hadamard finite-part integrals . Numer. Math. , 36 : 447 – 453 .
  • Sansone , G. 1959 . Orthogonal Functions , New York : Interscience publishers Inc. .
  • Shen , Y. J. and Lin , W. 2004 . The natural integral equations of plane elasticity problem and its wavelet methods . Appl. Math. Comput. , 150 : 417 – 438 .
  • Sun , W. and Wu , J. M. 2005 . Newton–Cotes formulae for the numerical evaluation of certain hypersingular integral . Computing , 75 : 297 – 309 .
  • Sun , W. and Wu , J. M. 2008 . Interpolatory quadrature rules for Hadamard finite-part integrals and their superconvergence . IMA J. Numer. Anal. , doi:10.1093/imanum/drm037; published online
  • Tsamasphyros , G. and Dimou , G. 1990 . Gauss quadrature rules for finite part integrals . Int. J. Numer. Methods Eng. , 30 : 13 – 26 .
  • Wahlbin , L. B. 1995 . Superconvergence in Galerkin Finite Element Methods , Springer . Lecture Notes in Math., 1605
  • Wu , J. M. and Lü , Y. 2005 . A superconvergence result for the second-order Newton–Cotes formula for certain finite-part integrals . IMA J. Numer. Anal. , 25 : 253 – 263 .
  • Wu , J. M. and Sun , W. 2005 . The superconvergence of the composite trapezoidal rule for Hadamard finite part integrals . Numer. Math. , 102 : 343 – 363 .
  • Wu , J. M. and Sun , W. 2008 . The superconvergence of Newton–Cotes rules for the Hadamard finite-part integral on an interval . Numer. Math. , 109 : 143 – 165 .
  • Wu , J. M. and Yu , D. H. 2000 . The natural boundary element method for exterior elliptic domain . Chinese J. Numer. Math. Appl. , 22 : 91 – 104 .
  • Wu , Z. P. , Kang , T. and Yu , D. H. 2004 . On the coupled NBEM and FEM for a class of nonlinear exterior Dirichlet problem in R2 . Sci. China Ser. A , 47 : 181 – 189 .
  • Yu , D. H. 1983 . Numerical solutions of harmonic and biharmonic canonical integral equations in interior or exterior circular domain . J. Comput. Math. , 21 : 52 – 62 .
  • Yu , D. H. 1985 . Approximation of boundary conditions at infinity for a harmonic equation . J. Comput. Math. , 3 : 219 – 227 .
  • Yu , D. H. 1991 . “ A direct and natural coupling of BEM and FEM ” . In Boundary Elements XIII , Edited by: Brebbia , C. A. and Gipson , G. S. 995 – 1004 . Southampton : Computational Mechanics Publications .
  • Yu , D. H. 1993 . The numerical computation of hypersingular integrals and its application in BEM . Adv. Eng. Software , 18 : 103 – 109 .
  • Yu , D. H. 1994 . The computation of hypersingular integrals on circle and its error estimates . Numer. Math.: J. Chin. Univ. , 16 : 332 – 337 .
  • Yu , D. H. 2002 . “ Natural boundary integral method and its applications ” . Kluwer Academic Publishers .
  • Yu , D. H. and Wu , J. M. 2001 . A nonoverlapping domain decomposition method for exterior 3-D problem . J. Comput. Math. , 19 : 77 – 86 .
  • Zhang , X. P. , Wu , J. M. and Yu , D. H. 2008 . Superconvergence of the composite Simpsons rule for a certain finite-part integral and its applications . J. Comput. Appl. Math. , doi:10.1016/j.cam.2008.02.001; published online

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