The multigrid  method for  a not  selfadjoint elliptical boundary
value problem that is a model of the Earth's ionosphere


  Dr. Valeri V. Denissenko
  Institute of Computational Modelling
  of the Russian Academy of Sciences
  Siberian Branch
  660036, Krasnoyarsk, Academgorodok, Russia
  denisen@cc.krascience.rssi.ru


                         ABSTRACT

As far as large scale electric fields and currents are concerned
the Earth's ionosphere may be simulated as two-dimensional
conductor. The proper mathematical model is a boundary value
problem for an elliptical equation. It's coefficients are equal
to the components of the conductivity tensor. This tensor is a
nonsymmetrical one due to the Hall effect. Values of coefficients
are huge in a thin near boundary strip.

To overcome difficulties that arise in numerical solution of
problems with not selfadjoint operators we replaced traditional
statements with new ones in which the operator is symmetrical and
positive definite.

Finite element equations are designed as the conditions of a
minimum of the energy functional as a function of values of
unknown piecewise linear approximating functions in grid points
and of parameters of special approximating functions that
separate near boundary singularity. We use regular nonuniform
grids.

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 equations solution.

We have 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 more general models of the Earth's
magnetosphere.