2016 , Volume 21, № 5, p.95-110

The paper proposed and implemented new versions of the method of collocation and least residuals (CLR) for the numerical solution of boundary value problems for PDE in domains with a curvilinear boundary. Their implementation and numerical experiments are performed on the examples of the biharmonic and Poisson equations. The solution of the biharmonic equation is used for simulation of the stress-strain state of an isotropic plate under the action of the transverse load. In the present study we apply the idea of using parts of the cells of a regular grid outside the domain, which are cut off by the boundary for the constructing the CLR methods. It is assumed that the solution has no singularities on the boundary and in a certain small neighborhood of it. The differential equation for the problem is true not only in the computational domain, but also in a small neighborhood of its boundary. The original domain with a curvilinear boundary is inscribed into a rectangle and then covered by a regular grid with rectangular cells, so that the points of fracture of boundary will be on the sides of the cells. We use the idea of joining the “small” irregular cells to the adjacent cells in order to reduce the condition number for the global system of linear algebraic equations. It is shown that the approximate solutions obtained by CLR converge with high order of accuracy thus accurately match the analytical solutions of test problems.