# Joint Applied Mathematics and Statistics Seminar 12.5. (Friday!)

Speaker: Alena Prakonina (Belarusian State University)

Title: ”Finite difference schemes and iterative methods for solving three-dimensional elliptic differential problems with mixed derivatives”

Abstract: ”Partial differential equations, in particular, elliptic-type equations have a wide range of applications, for example, in problems of heat conduction, diffusion, potential theory, etc. As a rule, in most practically significant cases it is not possible to find the exact analytical solution of such equations that is why in these cases approximate and numerical methods are more suitable and efficient to use. Numerical methods for solving differential equations are based on the construction of a discrete model of a differential problem in the form of systems of linear algebraic equations. The last ones later can be solved by both direct and iterative methods.

In this research work we consider the construction of difference schemes and iterative methods for solving three-dimensional elliptic problems with mixed derivatives and discontinuous coefficients. Particular interest was in finite-difference schemes for approximation of mixed derivatives, and there were considered 5 different stencils of the second order of accuracy. Achieved system of linear algebraic equations was successfully solved via using iterative technique, based on the method of bi-conjugate gradients with the spectrally optimal Fourier-Jacobi preconditioner, which made it possible to practically eliminate the growth of the number of iterations to achieve a given accuracy with a decrease in the grid step size and strong heterogeneity of the coefficients of the problem.

As test examples, three physical models were considered: the problem of anisotropic diffusion with smooth coefficients, the problem for the potential in a conducting anisotropic layered sphere and a thin anisotropic ring conductor in the environment of a weakly conducting isotropic medium. On the basis of numerical experiments it was found that the greatest stability of the convergence of iterative methods is demonstrated by schemes in which the templates for approximating the anisotropic component of the diffusion operator are consistent with the standard templates for isotropic problems (the averaging of the coefficients for mixed derivatives is performed on the same set of points of the template as in isotropic problem).

The work has great applications in forward and inverse EET/EIG problems.”

Time: 12:15
Place: Lecture Hall XVI (Quantum 1st floor)

All interested are very welcome!