Energy Method for the Elliptic Boundary Value Problems with Asymmetric Operators in a Spherical Layer

Valery V Denisenko,Institute Of Computational Modelling Sb Ras ,Semen A Nesterov

doi:10.17516/1997-1397-2021-14-5-554-565

Valery V Denisenko, Institute Of Computational Modelling Sb Ras + Show 1 more

Open Access

https://doi.org/10.17516/1997-1397-2021-14-5-554-565

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Abstract

Three-dimensional elliptic boundary value problems arising in the mathematical modeling of quasi-stationary electric fields and currents in conductors with gyrotropic conductivity tensor in domains homeomorphic to the spherical layer are considered. The same problems are mathematical models of thermal conductivity or diffusion in moving or gyrotropic media. The operators of the problems in the traditional formulation are non-symmetric. New statements of the problems with symmetric positive definite operators are proposed. For the four boundary value problems the quadratic energy functionals, to the minimization of which the solutions of these problems are reduced, are constructed. Estimates of the obtained quadratic forms are made in comparison with the form appearing in the Dirichlet principle for the Poisson equation

Highlights

The tree-dimensional elliptic boundary value problems with asymmetric operators arise, for example, in mathematical modeling of quasi-stationary electric fields and currents in the global conductor, consisting of the Earth’s ionosphere and atmosphere, using the domain decomposition method [1]
Three-dimensional elliptic boundary value problems arising in the mathematical modeling of quasi-stationary electric fields and currents in conductors with gyrotropic conductivity tensor in domains homeomorphic to the spherical layer are considered
Within the framework of the decomposition method [1], the boundary conditions arise at the selected boundaries of subdomains

Summary

Introduction

The tree-dimensional elliptic boundary value problems with asymmetric operators arise, for example, in mathematical modeling of quasi-stationary electric fields and currents in the global conductor, consisting of the Earth’s ionosphere and atmosphere, using the domain decomposition method [1]. In such models as [2], a significant simplification of the description of the D-layer of the Earth’s ionosphere, lying at heights of 50 − 90 km, is used. The corresponding estimates are obtained with specific values of the constants that will allow us to estimate the condition number of the matrix of a system of linear algebraic equations, which will arise in the numerical solution of the problem

The electric current continuity problem

The energy scalar product

The energy functional

The generalized solution