A spectral Galerkin method for a boundary integral equation

Author:
W. McLean

Journal:
Math. Comp. **47** (1986), 597-607

MSC:
Primary 65R20; Secondary 45L10

DOI:
https://doi.org/10.1090/S0025-5718-1986-0856705-2

MathSciNet review:
856705

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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.

**[1]**Douglas N. Arnold,*A spline-trigonometric Galerkin method and an exponentially convergent boundary integral method*, Math. Comp.**41**(1983), no. 164, 383–397. MR**717692**, https://doi.org/10.1090/S0025-5718-1983-0717692-8**[2]**J. Bergh & L. Löfström,*Interpolation Spaces*, Springer-Verlag, Berlin and New York, 1976.**[3]**Søren Christiansen,*On two methods for elimination of nonunique solutions of an integral equation with logarithmic kernel*, Applicable Anal.**13**(1982), no. 1, 1–18. MR**647662**, https://doi.org/10.1080/00036818208839372**[4]**R. E. Edwards,*Functional analysis. Theory and applications*, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR**0221256****[5]**Peter Henrici,*Fast Fourier methods in computational complex analysis*, SIAM Rev.**21**(1979), no. 4, 481–527. MR**545882**, https://doi.org/10.1137/1021093**[6]**G. C. Hsiao, P. Kopp, and W. L. Wendland,*A Galerkin collocation method for some integral equations of the first kind*, Computing**25**(1980), no. 2, 89–130 (English, with German summary). MR**620387**, https://doi.org/10.1007/BF02259638**[7]**George C. Hsiao and Wolfgang L. Wendland,*A finite element method for some integral equations of the first kind*, J. Math. Anal. Appl.**58**(1977), no. 3, 449–481. MR**0461963**, https://doi.org/10.1016/0022-247X(77)90186-X**[8]**M. A. Jaswon and G. T. Symm,*Integral equation methods in potential theory and elastostatics*, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1977. Computational Mathematics and Applications. MR**0499236****[9]**U. Lamp, K.-T. Schleicher, and W. L. Wendland,*The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations*, Numer. Math.**47**(1985), no. 1, 15–38. MR**797875**, https://doi.org/10.1007/BF01389873**[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*, Miniconference on linear analysis and function spaces (Canberra, 1984) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 9, Austral. Nat. Univ., Canberra, 1985, pp. 223–240. MR**825529****[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*, Translated from the Russian by R. S. Anderssen, Wolters-Noordhoff Publishing, Groningen, 1971. MR**0278506****[15]**S. M. Nikol′skiĭ,*Approximation of functions of several variables and imbedding theorems*, Springer-Verlag, New York-Heidelberg., 1975. Translated from the Russian by John M. Danskin, Jr.; Die Grundlehren der Mathematischen Wissenschaften, Band 205. MR**0374877****[16]**Gregory Verchota,*Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains*, J. Funct. Anal.**59**(1984), no. 3, 572–611. MR**769382**, https://doi.org/10.1016/0022-1236(84)90066-1**[17]**Rudolf Wegmann,*Convergence proofs and error estimates for an iterative method for conformal mapping*, Numer. Math.**44**(1984), no. 3, 435–461. MR**757498**, https://doi.org/10.1007/BF01405574

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DOI:
https://doi.org/10.1090/S0025-5718-1986-0856705-2

Article copyright:
© Copyright 1986
American Mathematical Society