A linear scheme for the numerical solution of nonlinear quasistationary magnetic fields

Author:
Miloš Zlámal

Journal:
Math. Comp. **41** (1983), 425-440

MSC:
Primary 65N30; Secondary 78-08, 78A30

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717694-1

MathSciNet review:
717694

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Abstract: The computation of nonlinear quasistationary two-dimensional magnetic fields leads to the following problem. There exists a bounded domain and an open nonempty set . We are looking for the magnetic vector potential which satisfies: (1) a certain nonlinear parabolic equation and an initial condition in *R*, (2) a nonlinear elliptic equation in , (3) a boundary conditon on and the condition that *u* as well as its conormal derivative are continuous across . This problem is formulated in an abstract variational way. We construct an approximate solution discretized in space by a generalized Galerkin method and by a one-step method in time. The resulting scheme is unconditionally stable and linear. A strong convergence of the approximate solution is proved without any regularity assumptions for the exact solution. We also derive an error bound for the solution of the two-dimensional magnetic field equations under the assumption that the exact solution is sufficiently smooth.

**[1]**J. Céa,*Optimization*, Dunod, Paris, 1971.**[2]**P. G. Ciarlet,*The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam, 1978. MR**0520174 (58:25001)****[3]**M. Crouzeix, "Une méthode multipas implicite-explicite pour l'approximation des équations paraboliques,"*Numer. Math.*, v. 35, 1980, pp. 257-276. MR**592157 (82b:65084)****[4]**J. Douglas, Jr. & T. Dupont, "Alternating-direction methods in rectangles," in*Numerical Solution of Partial Differential Equations*II, (B. Hubbard, ed). Academic Press, London and New York, 1971. MR**0273830 (42:8706)****[5]**A. Kufner, O. John & S. Fučik,*Function Spaces*, Academia, Prague, 1977.**[6]**J. L. Lions,*Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires*, Dunod, Gauthier-Villars, Paris, 1969. MR**0259693 (41:4326)****[7]**J. Nečas,*Les Méthodes Directes en Théorie des Equations Elliptiques*, Academia, Prague, 1967.**[8]**P. Temam,*Navier-Stokes Equations*, North-Holland, Amsterdam, 1977.**[9]**M. Zlámal, "Finite element solution of quasistationary nonlinear magnetic fields",*RAIRO Anal. Numer.*, v. 16, 1982, pp. 161-191. See also Addendum to the paper "Finite element solution of quasistationary nonlinear magnetic fields",*ibid.*, v. 17, 1983. MR**661454 (83k:65086)****[10]**N. A. Demerdash & D. H. Gillot, "A new approach for determination of eddy current and flux penetration in nonlinear ferromagnetic materials",*IEEE Trans. MAG*-10, 1974, pp. 682-685.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1983-0717694-1

Article copyright:
© Copyright 1983
American Mathematical Society