# Article

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Keywords:
finite element method; parabolic-elliptic problems; two-dimensional electromagnetic field
Summary:
The computation of nonlinear quasistationary two-dimensional magnetic fields leads to a nonlinear second order parabolic-elliptic initial-boundary value problem. Such a problem with a nonhomogeneous Dirichlet boundary condition on a part $\Gamma \!_1$ of the boundary is studied in this paper. The problem is discretized in space by the finite element method with linear functions on triangular elements and in time by the implicit-explicit method (the left-hand side by the implicit Euler method and the right-hand side by the explicit Euler method). The scheme we get is linear. The strong convergence of the method is proved under the assumptions that the boundary $\partial \Omega$ is piecewise of class $C^3$ and the initial condition belongs to $L_2$ only. Strong monotonicity and Lipschitz continuity of the form $a(v,w)$ is not an assumption, but a property of this form following from its physical background.
References:
[1] J. Céa: Optimisation. Dunod, Paris, 1971. MR 0298892
[2] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. MR 0520174 | Zbl 0383.65058
[3] M. Crouzeix: Une méthode multipas implicite-explicite pour l’approximation des équations d’évolution paraboliques. Numer. Math. 35 (1980), 257–276. DOI 10.1007/BF01396412 | MR 0592157 | Zbl 0419.65057
[4] 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), 682–685.
[5] J. Douglas, T.  Dupont: Alternating-direction Galerkin methods in rectangles. In: Proceedings 2nd Sympos. Numerical Solution of Partial Differential Equations II, Academic Press, London and New York, 1971, pp. 133–214. MR 0273830
[6] M. Feistauer, A.  Ženíšek: Finite element solution of nonlinear elliptic problems. Numer. Math. 50 (1987), 451–475. MR 0875168
[7] S. Fučík, A.  Kufner: Nonlinear Differential Equations. SNTL, Praha, 1978.
[8] D. Říhová-Škabrahová: A note to Friedrichs’ inequality. Arch. Math. 35 (1999), 317–327. MR 1744519
[9] A. Ženíšek: Curved triangular finite $C^m$-elements. Apl. Mat. 23 (1978), 346–377. MR 0502072
[10] A. Ženíšek: Approximations of parabolic variational inequalities. Appl. Math. 30 (1985), 11–35. MR 0779330
[11] A.  Ženíšek: Finite element variational crimes in parabolic-elliptic problems. Numer. Math. 55 (1989), 343–376. DOI 10.1007/BF01390058 | MR 0993476
[12] A. Ženíšek: Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations. Academic Press, London, 1990. MR 1086876
[13] M. Zlámal: Curved elements in the finite element method. I. SIAM J.  Numer. Anal. 10 (1973), 229–240. DOI 10.1137/0710022 | MR 0395263
[14] M.  Zlámal: Finite element solution of quasistationary nonlinear magnetic field. RAIRO Modél. Math. Anal. Numér. 16 (1982), 161–191. DOI 10.1051/m2an/1982160201611 | MR 0661454
[15] M.  Zlámal: A linear scheme for the numerical solution of nonlinear quasistationary magnetic fields. Math. Comp. 41 (1983), 425–440. DOI 10.2307/2007684 | MR 0717694

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