By Friedel Hartmann

to Boundary components thought and functions With 194 Figures Springer-Verlag Berlin Heidelberg manhattan London Paris Tokyo Hong Kong Dr.-Ing. Friedel Hartmann collage of Dortmund division of Civil Engineering 4600 Dortmund 50 FRG ISBN-13: 978-3-642-48875-7 e-ISBN-13: 978-3-642-48873-3 001: 10.1007/978-3-642-48873-3 Library of Congress Cataloging-in-Publication facts Hartmann, F. (Friedel) creation to boundary components: concept and applications/Friedel Hartmann. ISBN-13: 978-3-642-48875-7 1. Boundary price difficulties. I. identify. TA347.B69H371989 515.3'5--dc19 89-4160 This paintings is topic to copyright. All rights are reserved, no matter if the full or a part of the cloth is worried, in particular the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, copy on microfilms or in alternative ways, and garage in information banks. Duplication of this book or elements thereof is just authorized lower than the availability of the German Copyright legislations of September 9,1965, in its model of June 24,1985, and a copyright cost should always be paid. Violations fall below the prosecution act of the German Copyright legislations. © Springer-Verlag Berlin Heidelberg 1989 Softcover reprint of the hardcover 1 st version 1989 using registered names, emblems, and so forth. during this booklet doesn't mean, even within the absence of a selected assertion, that such names are exempt from the proper protecting legislation and laws and for this reason unfastened for common use.

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**Example text**

The optimal position of the collocation point depends on the principal part of the pseudo differential operator. A first kind integral equation in the plane with a Cauchy singular kernel gives poor results if we use naive collocation but the solution improves dramatically if we place the collocation points in between the boundary element nodes. Vice versa: a second kind integral equation with a smooth kernel gives good results with naive collocation and not so good results if we place the collocation points in between the nodes, [20].

The mathematical theory of integral equations is embedded into the modern Fourier analysis for pseudo differential operators. ) The decisive concept in this analysis is the concept of the strong ellipticity of an integral operator, [10]. If Garding's inequality is satisfied then this is equivalent to the positive definitness of the principal symbol of the pseudo differential operator, and since the GArding inequality implies stability and convergence of every Galerkin approximation this is particularly true for the boundary element Galerkin methods.

8 shows the range of admissible indices rand s. From this diagram we conclude that the highest possible order of convergence that can be achieved is collocation Galerkin least square O(hk-20') O(h 2k - 2o ) O(h2k-40') in H 2 o, in H 2 o- k in H 40'-k , The interesting point is, that these rates also determine the order of the error in the domain because the error in the interior IU - uhl = I! g(y,z)(f(y) - Ih(y))dsyl r ~ (collocation, note that 2a ell g[zlll-20' III - fhll20' < 0) can be estimated by the norm of the error on the boundary and the dual norm of the kernel.