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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
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 $ \Omega $ and an open nonempty set $ R \subset \Omega $. We are looking for the magnetic vector potential $ u({x_1},{x_2},t)$ which satisfies: (1) a certain nonlinear parabolic equation and an initial condition in R, (2) a nonlinear elliptic equation in $ S = \Omega - \bar R$, (3) a boundary conditon on $ \partial \Omega $ and the condition that u as well as its conormal derivative are continuous across $ \Gamma = \partial R \cap \partial S$. 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.

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Article copyright: © Copyright 1983 American Mathematical Society