# Article

 Title: Linear scheme for finite element solution of nonlinear parabolic-elliptic problems with nonhomogeneous Dirichlet boundary condition (English) Author: Říhová-Škabrahová, Dana Language: English Journal: Applications of Mathematics ISSN: 0862-7940 Volume: 46 Issue: 2 Year: 2001 Pages: 103-144 Summary lang: English . Category: math . 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. (English) Keyword: finite element method Keyword: parabolic-elliptic problems Keyword: two-dimensional electromagnetic field MSC: 35M10 MSC: 65M12 MSC: 65M60 MSC: 65N30 MSC: 78M10 idZBL: Zbl 1066.65117 idMR: MR1818081 DOI: 10.1023/A:1013783722140 . Date available: 2009-09-22T18:06:07Z Last updated: 2015-05-19 Stable URL: http://hdl.handle.net/10338.dmlcz/134460 . Reference: [1] J. Céa: Optimisation.Dunod, Paris, 1971. MR 0298892 Reference: [2] P. G. Ciarlet: The Finite Element Method for Elliptic Problems.North-Holland, Amsterdam, 1978. Zbl 0383.65058, MR 0520174 Reference: [3] M. Crouzeix: Une méthode multipas implicite-explicite pour l’approximation des équations d’évolution paraboliques.Numer. Math. 35 (1980), 257–276. Zbl 0419.65057, MR 0592157, 10.1007/BF01396412 Reference: [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. Reference: [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 Reference: [6] M. Feistauer, A.  Ženíšek: Finite element solution of nonlinear elliptic problems.Numer. Math. 50 (1987), 451–475. MR 0875168 Reference: [7] S. Fučík, A.  Kufner: Nonlinear Differential Equations.SNTL, Praha, 1978. Reference: [8] D. Říhová-Škabrahová: A note to Friedrichs’ inequality.Arch. Math. 35 (1999), 317–327. MR 1744519 Reference: [9] A. Ženíšek: Curved triangular finite $C^m$-elements.Apl. Mat. 23 (1978), 346–377. MR 0502072 Reference: [10] A. Ženíšek: Approximations of parabolic variational inequalities.Appl. Math. 30 (1985), 11–35. MR 0779330 Reference: [11] A.  Ženíšek: Finite element variational crimes in parabolic-elliptic problems.Numer. Math. 55 (1989), 343–376. MR 0993476, 10.1007/BF01390058 Reference: [12] A. Ženíšek: Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations.Academic Press, London, 1990. MR 1086876 Reference: [13] M. Zlámal: Curved elements in the finite element method.I. SIAM J.  Numer. Anal. 10 (1973), 229–240. MR 0395263, 10.1137/0710022 Reference: [14] M.  Zlámal: Finite element solution of quasistationary nonlinear magnetic field.RAIRO Modél. Math. Anal. Numér. 16 (1982), 161–191. MR 0661454 Reference: [15] M.  Zlámal: A linear scheme for the numerical solution of nonlinear quasistationary magnetic fields.Math. Comp. 41 (1983), 425–440. MR 0717694, 10.2307/2007684 .

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