Summary
This paper further develops the qualocation method for the solution of integral equations on smooth closed curves. Qualocation is a (Petrov-)Galerkin method in which the outer integrals are performed numerically by special quadrature rules. Here we allow the use of splines as trial and test functions, and prove stability and convergence results for certain qualocation rules when the underlying (Petrov-)Galerkin method is stable. In the common case of a first kind equation with the logarithmic kernel, a typical qualocation method employs piecewise-linear splines, and performs the outer integrals using just two points per interval. This method, with appropriate choice of the quadrature points and weights, has orderO(h 5) convergence in a suitable negative norm. Numerical experiments suggest that these rates of convergence are maintained for integral equations on open intervals when graded meshes are used to cope with unbounded solutions.
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Dedicated to Prof. Dr. Günther Hämmerlin on the occasion of his 60th birthday.
This author's work was supported by the A. von Humboldt-Stiftung
This author's work was supported by the Deutsche Forschungsgemeinschaft
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Chandler, G.A., Sloan, I.H. Spline qualocation methods for boundary integral equations. Numer. Math. 58, 537–567 (1990). https://doi.org/10.1007/BF01385639
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DOI: https://doi.org/10.1007/BF01385639