The mathematical model of an electric conductor is a boundary value problem for an elliptical equation. It's coefficients form a nonsymmetrical tensor because of Hall effect. We replaced traditional statement with new one in which the operator is symmetrical and positive definite.
Finite element equations are designed as the conditions of a minimum of the energy functional. The matrix of the system of linear algebraic equations is symmetrical and positive definite. It's condition number has the same dependence of grid step as one for the Poisson equation. We use a multigrid method to obtain the solution of the system.
As far as the Earth's ionosphere is concerned the main difficulty is due to huge values of coefficients in a thin strip near the boundary. It represents so called equatorial jet. We separate this singularity by a special boundary condition. It's approximation gives a subsystem with 5-diagonals symmetrical matrix in the system of finite element equations. The estimation of the approximation error for the boundary condition is of the same order as one for the equations inside the domain. In multigrid method we use special interpolation formulas for parameters at the boundary to keep the same structure of equations for all grids. The convergence rate of multigrid iterations is approximately the same as ones for boundary value problems with insulator or superconductor at the boundary.
We conducted numerical experiments with the designed model on the base of available data of satellite and ground based measurements. As a result the models of electric fields and currents distributions in the Earth's ionosphere are designed for substorms and for quiet geomagnetic conditions.
The designed mathematical model of the ionosphere as a global conductor is also using in our more general models of the Earth's magnetosphere. In these models the ionosphere is a passive object in that electric energy of magnetospheric generators dissipates.