A spline-trigonometric Galerkin method and an exponentially convergent boundary integral method
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- by Douglas N. Arnold PDF
- Math. Comp. 41 (1983), 383-397 Request permission
Abstract:
We consider a Galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions. We analyze the method applied to the integral equation of the first kind arising from a single layer potential formulation of the Dirichlet problem in the interior or exterior of an analytic plane curve. In constrast to ordinary spline Galerkin methods, we show that the method is stable, and so provides quasioptimal approximation, in a large family of Hilbert spaces including all the Sobolev spaces of negative order. As a consequence we prove that the approximate solution to the Dirichlet problem and all its derivatives converge pointwise with exponential rate.References
- Satya N. Atluri, Richard H. Gallagher, and O. C. Zienkiewicz (eds.), Hybrid and mixed finite element methods, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. Papers from the International Symposium in honor of Theodore Pian held at the Georgia Institute of Technology, Atlanta, Ga., April 8–10, 1981. MR 718335
- Douglas N. Arnold and Wolfgang L. Wendland, On the asymptotic convergence of collocation methods, Math. Comp. 41 (1983), no. 164, 349–381. MR 717691, DOI 10.1090/S0025-5718-1983-0717691-6
- A. K. Aziz and R. Bruce Kellogg, Finite element analysis of a scattering problem, Math. Comp. 37 (1981), no. 156, 261–272. MR 628694, DOI 10.1090/S0025-5718-1981-0628694-2
- Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI 10.1007/BF02165003
- Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106
- Gaetano Fichera, Linear elliptic equations of higher order in two independent variables and singular integral equations, with applications to anistropic inhomogeneous elasticity, Partial differential equations and continuum mechanics, Univ. Wisconsin Press, Madison, Wis., 1961, pp. 55–80. MR 0156084
- Hans-Peter Helfrich, Simultaneous approximation in negative norms of arbitrary order, RAIRO Anal. Numér. 15 (1981), no. 3, 231–235 (English, with French summary). MR 631677, DOI 10.1051/m2an/1981150302311
- Peter Henrici, Fast Fourier methods in computational complex analysis, SIAM Rev. 21 (1979), no. 4, 481–527. MR 545882, DOI 10.1137/1021093 G. Hsiao, On the Stability of Integral Equations of the First Kind with Logarithmic Kernels, Tech. Rep. No. 103-A, Appl. Math. Inst., Univ. of Delaware, 1981.
- 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, DOI 10.1007/BF02259638
- George Hsiao and R. C. MacCamy, Solution of boundary value problems by integral equations of the first kind, SIAM Rev. 15 (1973), 687–705. MR 324242, DOI 10.1137/1015093
- 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 461963, DOI 10.1016/0022-247X(77)90186-X
- G. C. Hsiao and W. L. Wendland, The Aubin-Nitsche lemma for integral equations, J. Integral Equations 3 (1981), no. 4, 299–315. MR 634453
- M. N. Le Roux, Méthode d’éléments finis pour la résolution numérique de problèmes extérieurs en dimension $2$, RAIRO Anal. Numér. 11 (1977), no. 1, 27–60, 112 (French, with English summary). MR 448954, DOI 10.1051/m2an/1977110100271
- I. J. Schoenberg, Cardinal interpolation and spline functions, J. Approximation Theory 2 (1969), 167–206. MR 257616, DOI 10.1016/0021-9045(69)90040-9
- Larry L. Schumaker, Spline functions: basic theory, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. MR 606200
- Gilbert Strang and George J. Fix, An analysis of the finite element method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. MR 0443377
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Math. Comp. 41 (1983), 383-397
- MSC: Primary 65N15; Secondary 41A15, 45L05, 65D07, 65R20
- DOI: https://doi.org/10.1090/S0025-5718-1983-0717692-8
- MathSciNet review: 717692