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finite element method; parabolic-elliptic problems; two-dimensional electromagnetic field
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.
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