A linear scheme for the numerical solution of nonlinear quasistationary magnetic fields

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.