A spline-trigonometric Galerkin method and an exponentially convergent boundary integral method

Author:
Douglas N. Arnold

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

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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.

**[1]**D. Arnold, I. Babuška & J. Osborn, "Selection of finite element methods,"*Proc. Internat. Sympos. on Hybrid and Mixed Finite Element Methods*(S. N. Atluri and R. H. Gallagher, eds.), Wiley, New York, 1981. MR**718335 (84i:73001)****[2]**D. Arnold & W. Wendland, "On the asymptotic convergence of collocation methods,"*Math. Comp.*. v. 41, 1983, pp. MR**717691 (85h:65254)****[3]**A. Aziz & B. Kellogg, "Finite element analysis of a scattering problem,"*Math. Comp.*, v. 37, 1981, pp. 261-272. MR**628694 (82i:65069)****[4]**I. Babuška, "Error-bounds for finite element method,"*Numer. Math.*, v. 16, 1970, pp. 322-333. MR**0288971 (44:6166)****[5]**I. Babuška & 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*(A. K. Aziz, ed.), Academic Press, New York, 1972. MR**0421106 (54:9111)****[6]**G. Fichera, "Linear elliptic equations of higher order in two independent variables and singular integral equations,"*Proc. Conf. on Partial Differential Equations and Continuum Mechanics*(Madison, Wise.), Univ. of Wisconsin Press, 1961. MR**0156084 (27:6016)****[7]**H.-P. Helfrich, "Simultaneous approximation in negative norms of arbitrary order,"*RAIRO Anal. Numér.*, v. 15, 1981, pp. 231-235. MR**631677 (82k:65054)****[8]**P. Henrici, "Fast Fourier methods in computational complex analysis,"*SIAM Rev.*, v. 15, 1979, pp. 481-527. MR**545882 (80i:65031)****[9]**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.**[10]**G. Hsiao, P. Kopp & W. Wendland, "A Galerkin collocation method for some integral equations of the first kind,"*Computing*, v. 25, 1980, pp. 89-130. MR**620387 (83e:65210)****[11]**G. Hsiao & R. McCamy, "Solution of boundary value problems by integral equations of the first kind,"*SIAM Rev.*, v. 15, 1973, pp. 687-705. MR**0324242 (48:2594)****[12]**G. Hsiao & W. 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)****[13]**G. Hsiao & W. Wendland, "The Aubin-Nitsche lemma for integral equations,"*J. Integral Equations*, v. 3, 1981, pp. 299-315. MR**634453 (83j:45019)****[14]**M. N. Le Roux, "Méthode d'éléments finis pour la résolution numérique de problèmes extérieurs en dimension 2,"*RAIRO Anal. Numér.*, v. 11, 1977, pp. 27-60. MR**0448954 (56:7259)****[15]**I. Schoenberg, "Cardinal interpolation and spline functions,"*J. Approx. Theory*, v. 2, 1969, pp. 167-206. MR**0257616 (41:2266)****[16]**L. Schumaker,*Spline Functions*:*Basic Theory*, Wiley, New York, 1981. MR**606200 (82j:41001)****[17]**G. Strang & G. Fix,*An Analysis of the Finite Element Method*, Prentice-Hall, Englewood Cliffs, N.J., 1973. MR**0443377 (56:1747)**

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DOI:
https://doi.org/10.1090/S0025-5718-1983-0717692-8

Article copyright:
© Copyright 1983
American Mathematical Society