Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A spectral Galerkin method for a boundary integral equation

Author: W. McLean
Journal: Math. Comp. 47 (1986), 597-607
MSC: Primary 65R20; Secondary 45L10
MathSciNet review: 856705
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the boundary integral equation which arises when the Dirichlet problem in two dimensions is solved using a single-layer potential. A spectral Galerkin method is analyzed, suitable for the case of a smooth domain and smooth boundary data. The use of trigonometric polynomials rather than splines leads to fast convergence in Sobolev spaces of every order. As a result, there is rapid convergence of the approximate solution to the Dirichlet problem and all its derivatives uniformly up to the boundary.

References [Enhancements On Off] (What's this?)

  • [1] D. N. Arnold, "A spline-trigonometric Galerkin method and an exponentially convergent boundary integral method," Math. Comp., v. 41, 1983, pp. 383-397. MR 717692 (84m:65117)
  • [2] J. Bergh & L. Löfström, Interpolation Spaces, Springer-Verlag, Berlin and New York, 1976.
  • [3] S. Christiansen, "On two methods for elimination of non-unique solutions of an integral equation with logarithmic kernel," Applicable Anal., v. 13, 1982, pp. 1-18. MR 647662 (83e:65204)
  • [4] R. E. Edwards, Functional Analysis, Holt, Rinehart & Winston, New York, 1965. MR 0221256 (36:4308)
  • [5] P. Henrici, "Fast Fourier methods in computational complex analysis," SIAM Rev., v. 21, 1979, pp. 481-527. MR 545882 (80i:65031)
  • [6] G. C. Hsiao, P. Kopp & W. L. Wendland, "A Galerkin collocation method for some integral equations of the first kind," Computing, v. 25, 1980, pp. 89-130. MR 620387 (83e:65210)
  • [7] G. C. Hsiao & W. L. Wendland, "A finite element method for some integral equations of the first kind," J. Math. Anal. Appl., v. 58, 1977, pp. 449-481. MR 0461963 (57:1945)
  • [8] M. A. Jaswon & G. T. Symm, Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, New York, 1977. MR 0499236 (58:17147)
  • [9] U. Lamp, K.-T. Schleicher & W. L. Wendland, "The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations," Numer. Math., v. 47, 1985, pp. 15-38. MR 797875 (86m:65145)
  • [10] W. McLean, Boundary Integral Methods for the Laplace Equation, Thesis, Australian National University, Canberra, 1985.
  • [11] W. McLean, A Computational Method for Solving a First Kind Integral Equation, Research Report CMA-R15-85, Centre for Mathematical Analysis, Australian National University, 1985.
  • [12] W. McLean, "Error estimates for a first kind integral equation and an associated boundary value problem," Proc. Centre Math. Anal. Austral. Nat. Univ., v. 9, 1985, pp. 223-240. MR 825529 (87e:65066)
  • [13] J. Marcinkiewicz, "Sur les multiplicateurs des séries de Fourier," Studia Math., v. 8, 1939, pp. 78-91.
  • [14] S. G. Mikhlin, The Numerical Performance of Variational Methods, Wolters-Noordhoff, Groningen, 1971. MR 0278506 (43:4236)
  • [15] S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, Berlin and New York, 1975. MR 0374877 (51:11073)
  • [16] G. Verchota, "Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains," J. Funct. Anal., v. 59, 1984, pp. 572-611. MR 769382 (86e:35038)
  • [17] R. Wegmann, "Convergence proofs and error estimates for an iterative method for conformal mapping," Numer. Math., v. 44, 1984, pp. 435-461. MR 757498 (85m:30004)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65R20, 45L10

Retrieve articles in all journals with MSC: 65R20, 45L10

Additional Information

Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society