Statements of the boundary value problems in mathematical simulation of a quasistationary electric field in the atmosphere and ionosphere

Abstract

Boundary-value problems which have to be solved in the frame of the mathematical models of quasi-stationary electric fields and currents in the global conductor, that consists of the Earth's ionosphere and the atmosphere are formulated. The approach based on the domain decomposition is used. A quadratic energy functional is constructed. It allows us to reduce the solution of the boundary value problem for a three-dimensional elliptic differential equation with asymmetric coefficients to minimizing of the functional. Estimates of the quadratic form in comparison with the Dirichlet principle for the Poisson equation are obtained. These estimates allow us to assess the condition number of the matrix of the system of linear algebraic equations, arising in the numerical solution to the problem.