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

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.

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

Article copyright:
© Copyright 1983
American Mathematical Society