FINITE ELEMENT METHOD FOR THREE-DIMENSIONAL CURRENT CONTINUITY PROBLEMS IN GYROTROPIC MEDIA // BSU Bulletin. Mathematics, Informatics. - 2022. №4. . - С. 12-29.

FINITE ELEMENT METHOD FOR THREE-DIMENSIONAL CURRENT CONTINUITY PROBLEMS IN GYROTROPIC MEDIA

Исследование выполнено за счет гранта Российского научного фонда № 22- 27-00006, https://rscf.ru/project/22-27-00006/

DOI: 10.18101/2304-5728-2022-4-12-29 UDK: 517.9

For a gyrotropic medium, the operators of elliptical bound- ary value current continuity problems, traditionally formulated for electric potential, are asymmetric, which makes it difficult to solve such problems numerically. In this paper, the formulation of the boundary value problem with a symmetric positive definite operator proposed by the author is used. New unknown functions are the pairs of special potentials, scalar and vector, which in particular cases coincide with the electric potential and the current function. Similar problems are formulated when modeling thermal conductivity or diffusion in moving or gyrotropic media. For the new problem, the principle of the minimum of the quadratic energy functional is valid, similar to the Dirichlet principle for the Poisson equation. The introduced energy norm is equivalent to the sum of the energy norms of four new unknown functions as elements of spaces used for basic boundary value problems for the Poisson equation. This makes it possible to use classical theorems of ap- proximation of solutions by piecewise linear functions and to create a simple finite element method, that is, to reduce the problem to a system of linear algebraic equations for nodal values of approximating functions. The matrix of this system is symmetric and positively defined. In this paper, the for- mulas necessary for constructing the coefficients of this matrix are derived, starting with geometric constructions in grid tetrahedra. A discrete model is proposed that allows interpreting one of the equations of the finite element method as the charge conservation law, integrated over the grid cells. The examples show the convergence of the resulting approximate solutions when the grid is fined.

elliptical equation, boundary value problem, electrical conductivity, gyrotropic medium, asymmetric operator, energy method, finite element method.