The paper is devoted to the issues of justification and implementation of the Galerkin method for solving initial boundary value problems for the system of Maxwell equations in the quasi-stationary electrical approximation that arise when modeling a global electric circuit in the atmosphere. Two variants of non-classical boundary conditions for the scalar electric potential at the conventional boundary between the atmosphere and the ionosphere are considered. The first type of boundary conditions is that the electric potential does not depend on spatial coordinates. Another variant of boundary conditions is based on the assumption of equipotentiality of the Earth’s magnetic field lines in the ionosphere and is a special type of conditions at magnetic conjugate points. The results on the well-posedness of the formulations of the problems under consideration are presented. The applicability of the Galerkin method for the quasi-stationary electric approximation with boundary conditions at magnetic conjugate points on the boundary with the ionosphere is strictly justified. Features of the finite element implementation of the method are discussed. A comparative analysis of solutions to problems with different boundary conditions is carried out.