Preprint
An approximation method using approximate approximations
Abstract
The aim of this paper is to extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the Laplace equation in two dimensions using approximate approximations. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.
Citation
@article{urn:nbn:de:hebis:34-200604069104,
author={Müller, Frank and Varnhorn, Werner},
title={An approximation method using approximate approximations},
year={2005}
}
0500 Oax 0501 Text $btxt$2rdacontent 0502 Computermedien $bc$2rdacarrier 1100 2005$n2005 1500 1/eng 2050 ##0##urn:nbn:de:hebis:34-200604069104 3000 Müller, Frank 3010 Varnhorn, Werner 4000 An approximation method using approximate approximations / Müller, Frank 4030 4060 Online-Ressource 4085 ##0##=u http://nbn-resolving.de/urn:nbn:de:hebis:34-200604069104=x R 4204 \$dPreprint 4170 Mathematische Schriften Kassel 7136 ##0##urn:nbn:de:hebis:34-200604069104
2006-04-06T11:39:17Z 2006-04-06T11:39:17Z 2005 urn:nbn:de:hebis:34-200604069104 http://hdl.handle.net/123456789/200604069104 132946 bytes application/pdf eng Universität Kassel, FB 17, Mathematik/Informatik Urheberrechtlich geschützt https://rightsstatements.org/page/InC/1.0/ Approximate approximations Laplace equation Dirichlet problem Boundary integral equation Approximationstheorie Laplace-Gleichung Dirichlet-Problem Integralgleichung 510 An approximation method using approximate approximations Preprint The aim of this paper is to extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the Laplace equation in two dimensions using approximate approximations. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given. open access Müller, Frank Varnhorn, Werner Mathematische Schriften Kassel 05, 11 65N38 45B05 35J05 Mathematische Schriften Kassel 05, 11
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