Geodesic
Linear scheme for finite element solution of nonlinear parabolic-elliptic problems with nonhomogeneous Dirichlet boundary condition
Applications of Mathematics, Tome 46 (2001) no. 2, pp. 103-144.

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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.
DOI : 10.1023/A:1013783722140
Classification : 35M10, 65M12, 65M60, 65N30, 78M10
Keywords: finite element method; parabolic-elliptic problems; two-dimensional electromagnetic field 
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     title = {Linear scheme for finite element solution of nonlinear parabolic-elliptic problems with nonhomogeneous {Dirichlet} boundary condition},
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Říhová-Škabrahová, Dana. Linear scheme for finite element solution of nonlinear parabolic-elliptic problems with nonhomogeneous Dirichlet boundary condition. Applications of Mathematics, Tome 46 (2001) no. 2, pp. 103-144. doi : 10.1023/A:1013783722140. http://geodesic.mathdoc.fr/articles/10.1023/A:1013783722140/

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