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The convergence of spline collocation for strongly elliptic equations on curves

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Summary

Most boundary element methods for two-dimensional boundary value problems are based on point collocation on the boundary and the use of splines as trial functions. Here we present a unified asymptotic error analysis for even as well as for odd degree splines subordinate to uniform or smoothly graded meshes and prove asymptotic convergence of optimal order. The equations are collocated at the breakpoints for odd degree and the internodal midpoints for even degree splines. The crucial assumption for the generalized boundary integral and integro-differential operators is strong ellipticity. Our analysis is based on simple Fourier expansions. In particular, we extend results by J. Saranen and W.L. Wendland from constant to variable coefficient equations. Our results include the first convergence proof of midpoint collocation with piecewise constant functions, i.e., the panel method for solving systems of Cauchy singular integral equations.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, 20742, College Park, MD, USA

    Douglas N. Arnold

  2. Fachbereich Mathematik, Technische Hochschule, D-6100, Darmstadt, Federal Republic of Germany

    Wolfgang L. Wendland

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  1. Douglas N. Arnold
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  2. Wolfgang L. Wendland
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Dedicated to Prof. Dr. Dr. h.c. mult. Lothar Collatz on the occasion of his 75th birthday

This work was begun at the Technische Hochschule Darmstadt where Professor Arnold was supported by a North Atlantic Treaty Organization Postdoctoral Fellowship. The work of Professor Arnold is supported by NSF grant BMS-8313247. The work of Professor Wendland was supported by the “Stiftung Volkswagenwerk”

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Arnold, D.N., Wendland, W.L. The convergence of spline collocation for strongly elliptic equations on curves. Numer. Math. 47, 317–341 (1985). https://doi.org/10.1007/BF01389582

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