Multigrid Method for a Global Hall Conductor in the Earth's Ionosphere

Valeri V. Denissenko
Institute of Computational Modelling
Krasnoyarsk, 660036, Russia

Abstract

Since conductivity is an asymmetric tensor in the ionosphere the elliptical operators of the boundary value problems are not selfadjoint. To overcome difficulties that arise in numerical solution of problems with not selfadjoint operators it is proposed to replace the traditional statements with new ones in which the operators are symmetrical and positive definite. A narrow boundary strip in that the coefficients are huge in comparing with those in the interior of the domain is separated with a special boundary condition. The finite element equations are obtained as a conditions of a minimum of the energy functional with a special set of parameters to approximate the main boundary conditions. Positive definiteness of the matrix is proved. A multigrid method that is used to solve these equations and a numerical example are presented.

Key words. gyrotropic transfer, Hall conductivity, symmetrization, elliptic equation, variational principle, finite elements, multigrid method

AMS subject classifications. Primary 65N30, 35J50; Secondary 78A25