A Backward Euler-DtN Alternating Method for Time-dependent Eddy Current Problems in Unbounded Domains

In this paper, we develop a coupling of natural boundary element method and finite element method for solving the time-dependent eddy current problems in unbounded domains based on a H−ψ formulation. The H−ψ formulation allows that a scalar function is used outside the conduc-tor and a vector function is only used in the conducting domain. To get a full discrete scheme of the problem, the backward Euler method is applied for the discretization of the time variable and the Nédélec element of the lowest order, the piecewise linear element and the curvilinear element are used for the discretization of the space variables. A backward Euler-DtN alternating method is designed to solve the discrete coupled problem. In each time step of the method, the action of the boundary operators can be implemented by calculating a series of the spherical harmonics, instead of solving boundary integral equations. We derive an error estimate of the fully discrete scheme, and prove the convergence of the alternating method. Numerical experiments show the effectiveness of the new method