On numerical methods for time-dependent eddy current problems for the Maxwell equations in inhomogeneous media

E-based and H-based formulations for time-dependent eddy current problems for the Maxwell equations in inhomogeneous media have been derived. The concept of generalized solutions for boundary-value problems in bounded regions for obtained systems of equations has been formulated. Conditions for the existence and uniqueness of the generalized solutions have been established. Axisymmetric problems have been thoroughly considered, and a class of test problems has been proposed. Their exact solutions have the same key features as the solutions of the original problems. Numerical methods based on finite element approximations of the three-dimensional operators have been constructed. Particular attention has been paid to the methods on tetrahedral elements. Linear Lagrange elements, zero-order and first-order Nédélec elements have been used. The computational efficiency of the proposed finite element approximations has been analyzed by solving the constructed test problems. For small gaps in the coefficients of equations and regular finite element meshes, the first-order Nédélec elements have certain advantages in terms of accuracy and computational costs.