Volume 4, Issue 1

In this work we consider the numerical solution of the Laplace's equation in a domain with holes by means of the overlapping of finite and boundary elements. The essence of the method is the consideration of the finite element solution of the Laplace's equation in the domain without holes and the exterior single{layer solution on the unbounded domain around these holes. This solution can be viewed as a limit of a discretized interior{exterior Schwarz-type iteration. A convergence analysis of both the iteration and the discrete solution is carried out, taking full generality in the BEM scheme. Some numerical experiments are also given.

In this paper we investigate two iterative methods for solving one problem of nonlinear optics. The main goal is not only to find a stationary solution but also to investigate its stability. It is shown that both methods have very different stability properties and the less stable algorithm is close to the approximation of the physically important non-stationary problem. We also propose a new iterative algorithm for solving a more complicated problem which describes the optical conjugation in stimulated Brillouin backscattering with pump depletion. This algorithm is based on a symmetrical splitting scheme and the nonlinear interaction is approximated by using the special mass conservation property of the discrete problem. Thus, we obtain a conservative iterative algorithm. The results of the numerical experiments are presented and they confirm our theoretical conclusions.

Energy estimates and convergence analysis of finite difference methods for Biot's consolidation model are presented for several types of radial ow. The model is written by a system of partial differential equations which depend on an integer parameter ( n = 0; 1; 2 ) corresponding to the one-dimensional ow through a deformable slab and the radial ow through an elastic cylindrical or spherical shell respectively. The finite difference discretization is performed on staggered grids using separated points for the approximation of pressure and displacements. Numerical results are given to illustrate the obtained theoretical results.

A survey of the results concerning the convergence of finite difference schemes for boundary value problems with generalized solutions from the Sobolev space is presented. In particular, difference schemes for some problems with singular coefficients are investigated.

Certain methods for numerical solving plane and axially symmetric problems on equilibrium shapes of a capillary surface are presented. The methods possess a high order of approximation on a nonuniform grid. They are easy to realize, fairly universal and suitable for constructing not only simply connected but also doubly connected and disconnected surfaces, including strongly curved ones. It is shown that the iterative algorithms constructed are absolutely stable at each iteration. The condition for convergence of iterations is obtained within the framework of a linear theory. To describe peak-shaped configurations of a magnetic uid in a high magnetic field, an algorithm of generation of adaptive grid nodes in accordance with the surface curvature is proposed. The methods have been tested for the well-known problems of capillary hydrostatics on equilibrium shapes of a drop adjacent to the horizontal rotating plate under gravity, and of an isolated magneticuid drop in a high uniform magnetic field. It has been established that they adequately respond to the physical phenomenon of a crisis of equilibrium shapes, i.e., they can be adopted to investigate the stability of equilibrium states of a capillary surface.

Certain schemes for approximate calculation of singular integrals with a Cauchy kernel and their application to the numerical solution of the modified Dirichlet problem are offered. Questions of justifying the corresponding computational schemes for domains with Lyapunov boundaries are investigated.

We study numerical approximations for a class of singularly perturbed convection-diffusion type problems with a moving interior layer. In a domain (segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not converge ε-uniformly in the uniform norm. In the case of rectangular meshes which are (a priori or a posteriori ) locally condensed in the transition layer. However, the condition for convergence can be considerably weakened if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an a priori, or an a posteriori adaptive mesh technique. Here we construct a scheme on a posteriori adaptive meshes (based on the solution gradient), whose solution converges ‘almost ε-uniformly’.