Spectral properties for the magnetization integral operator
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 by Mark J. Friedman and Joseph E. Pasciak PDF
 Math. Comp. 43 (1984), 447453 Request permission
Abstract:
We analyze the spectrum of a certain singular integral operator on the space ${({L^2}(\Omega ))^3}$ where $\Omega$ is contained in three dimensional Euclidean space and has a Lipschitz continuous boundary. This operator arises in the integral formulation of the magnetostatic field problem. We decompose ${({L^2}(\Omega ))^3}$ into invariant subspaces: in one where the operator is the zero map; in one, the identity map; and in one where the operator is positive definite and bounded. These results give rise to the formulation of new efficient numerical techniques for approximating nonlinear magnetostatic field problems [5], [6], [12].References

A. G. Armstrong, A. M. Collie, C. J. Diserens, N. J. Newman, M. Simkin & C. W. Trowbridge, New Developments in the Magnet Design Program GFUN, Rutherford Laboratory Report No. RL5060.
 Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, NorthHolland Publishing Co., AmsterdamNew YorkOxford, 1978. MR 0520174
 Mark J. Friedman, Mathematical study of the nonlinear singular integral magnetic field equation. I, SIAM J. Appl. Math. 39 (1980), no. 1, 14–20. MR 585825, DOI 10.1137/0139003
 Mark J. Friedman, Mathematical study of the nonlinear singular integral magnetic field equation. II, SIAM J. Numer. Anal. 18 (1981), no. 4, 644–653. MR 622700, DOI 10.1137/0718042 M. J. Friedman & J. S. Colonias, "On the coupled differentialintegral equations for the solution of the general magnetostatic problem," IEEE Trans. Mag., v. Mag18, No. 2, March 1982, pp. 336339.
 Mark J. Friedman, Finite element formulation of the general magnetostatic problem in the space of solenoidal vector functions, Math. Comp. 43 (1984), no. 168, 415–431. MR 758191, DOI 10.1090/S00255718198407581918 J. L. Lions & E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. 1, SpringerVerlag, New York, 1972.
 S. G. Mikhlin, Multidimensional singular integrals and integral equations, Pergamon Press, OxfordNew YorkParis, 1965. Translated from the Russian by W. J. A. Whyte; Translation edited by I. N. Sneddon. MR 0185399 J. Nečas, Les Méthodes Directes en Théorie des Équations Elliptiques, Masson, Paris, 1967.
 J.C. Nédélec, Homogénéisation du problème des courants de Foucault dans un transformateur, Séminaire sur les Équations aux Dérivées Partielles 1987–1988, École Polytech., Palaiseau, 1988, pp. Exp. No. XV, 10 (French). MR 1018187
 Joseph E. Pasciak, An iterative algorithm for the volume integral method for magnetostatics problems, Comput. Math. Appl. 8 (1982), no. 4, 283–290. MR 679401, DOI 10.1016/08981221(82)900104
 Joseph E. Pasciak, A new scalar potential formulation of the magnetostatic field problem, Math. Comp. 43 (1984), no. 168, 433–445. MR 758192, DOI 10.1090/S0025571819840758192X
 Roger Temam, NavierStokes equations, 3rd ed., Studies in Mathematics and its Applications, vol. 2, NorthHolland Publishing Co., Amsterdam, 1984. Theory and numerical analysis; With an appendix by F. Thomasset. MR 769654
 V. S. Vladimirov, Equations of mathematical physics, Pure and Applied Mathematics, vol. 3, Marcel Dekker, Inc., New York, 1971. Translated from the Russian by Audrey Littlewood; Edited by Alan Jeffrey. MR 0268497
Additional Information
 © Copyright 1984 American Mathematical Society
 Journal: Math. Comp. 43 (1984), 447453
 MSC: Primary 78A30; Secondary 47G05
 DOI: https://doi.org/10.1090/S00255718198407581931
 MathSciNet review: 758193