Abstract
Boundary collocation is a method for computing solutions to boundary problems for linear partial differential equations, for which complete families of particular solutions are explicitly known. The method is simple to implement on a computer and has been found to be competitive for many problems of electrical engineering.
An application of boundary collocation requires several choices, such as the choices of subspace and basis of particular solutions, and the choice of collocation points. The numerical aspects of these choices have so far only received little attention. This paper investigates how subspace, basis and collocation points should be chosen when solving a model interface problem. The purpose of this investigation is to obtain guidelines for these choices applicable also to other boundary problems. These guidelines have been used to solve two problems that arose in Swedish industry: an investigation of microwave heating of food with the aim of designing equipment to achieve uniform heating, and a study of the heat conduction of a rock important for the design of a hot water storage near Uppsala for water heated by solar collectors.
Research supported in part by NSF under Grant DMS-8704196.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abramowitz, M; Stegun, I.E., eds.: Handbook of Mathematical Functions. National Bureau of Standards 1972.
Bates, R.H.T.: Analytic constraints on electromagnetic field computations. IEEE Trans. Microwave Theory Tech. MTT-23 (1975) 608–623.
Bates, R.H.T.; Ng, F.L.: Point matching computation of transverse resonances. Int’l J. Numer. Meth. Engng. 6 (1973) 155–168.
Bieberbach, L: Conformal Mapping. New York: Chelsea 1964.
Davis, P.J.: The Schwarz Function and its Applications. Math. Assoc. Amer. 1974.
Gautschi, W.: Conditions of polynomials in power form. Math. Comput. 33 (1979) 343–352.
Jones, D.S.: Methods in Electromagnetic Wave Propagation, vol. 1. Oxford: Oxford University Press 1987.
Ohlsson, T.; Risman, P.O.: Temperature distribution of microwave heating — spheres and cylinders. J. Microwave Power 13 (1978) 303–310.
Rehbinder, G.; Reichel, L.: Heat conduction in a rock mass with an annular hot water storage. Int’l J. Heat and Fluid Flow 5 (1984) 131–137.
Reichel, L.: On the determination of boundary collocation points for solving some problems for the Laplace operator. J. Comput. Appl. Math. 11 (1984) 175–196.
Reichel, L.: On the numerical solution of some 2-d electromagnetic interface problems by the boundary collocation method. Comput. Math. Appl. Mech. Engng. 53 (1985) 1–11.
Reichel, L.: Numerical methods for analytic continuation and mesh generation. Constr. Approx. 2 (1986) 23–39.
Reichel, L.: Some computational aspects of a method for rational approximation. SIAM J. Sci. Stat. Comput. 7 (1986) 1041–1057.
Reichel, L.: A fast method for solving certain integral equations with application to conformal mapping. J. Comput. Appl. Math. 14 (1986) 125–142.
Walsh, J.L.: Interpolation and Approximation by Rational Functions in the Complex Domain, 5th ed. Providence: Amer. Math. Society 1969.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 B. G. Teubner Stuttgart and Kluwer Academic Publishers
About this chapter
Cite this chapter
Reichel, L. (1990). Solving a Model Interface Problem for the Laplace Operator by Boundary Collocation and Applications. In: Manley, J., McKee, S., Owens, D. (eds) Proceedings of the Third European Conference on Mathematics in Industry. European Consortium for Mathematics in Industry, vol 5. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0629-7_11
Download citation
DOI: https://doi.org/10.1007/978-94-009-0629-7_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6770-6
Online ISBN: 978-94-009-0629-7
eBook Packages: Springer Book Archive