# collocation method

## Scattering of plane waves by an open arc with different boundary conditions on its sides

The paper considers an approach to  numerical modelling of the problem of electromagnetic plane waves scattering by an open arc with different boundary conditions on its sides. The corresponding mixed boundary value problem is reduced to a system of two singular and hypersingular integral equations. The method of collocation for numerical solutions to the equations obtained is proposed and analyzed. Numerical results of the radar cross-section for different values of arc curvatures are presented.

## Finite element approximations in projection methods for solution of some Fredholm integral equation of the first kind

Approximation properties of B-splines and Lagrangian finite elements in Hilbert spaces of functions defined on surfaces in three-dimensional space are investigated.  The conditions for the convergence of Galerkin and collocation methods for solution of the Fredholm integral equation of the first kind for the simple layer potential that is equivalent to the Dirichlet problem for Laplace equation in $\mathbb{R}^3$ are established.  The estimation of the error of approximate solution of this problem, obtained by means of the potential theory methods, is determined.

## Fourier method employed for solving thermoelastisity problems for rectangular plate with opening and inclusions

Dynamic problems of thermoelasticity for the rectangular plate with absolutely rigid inclusion and opening are considered. The problem is reduced to the system of boundary integral equations, numerical solution of which is constructed by the collocation method.

## Estimation of the Earth crust strain rate tensor from GPS observations data in the Eastern Europe

Estimated from GPS observations velocities of GPS-stations were used to obtain 2D-model velocities and strain rate field in the Eastern Europe. The study of the velocities field in the region was done in a few steps. The first one consists of the development of the finite element approach on the geosphere based on bicubic spline functions and least squares collocation method for the interpolation scattered GPS-data to the regular nodes. The second one represents the inversion of velocities from GPS-observations to the strain rate tensor.

## The field of linear velocities and movements of the earth crust in the South Eastern Europe region

Coordinates and velocities of GPS-points in the South-East Europe were used to derive the 2D model of velocities and horizontal strain rate fields of the region (the territory of Ukraine, Bulgaria, Romania, Greece, Turkey and Iran). The first step of studies consists of the development of the finite element approach based on the bicubic spline functions for the building of the regular grid data via scattered geodetic network. The second one represents the inversion of velocities from GPS observations data to the strain rate tensor.