Band 4, Heft 3

In this article we prove the existence of two different classes of regular hexagons in the hexagonal grid. We develop a generalized layer-wise construction of a hexagonal discrete velocity model and derive general formulae to identify all regular hexagons belonging to the grid. We also present some numerical results based on the hexagonal grid.

We adapt, via asymptotic expansion, Kapitsa's formula for the effective potential of a pendulum with vibrating suspension to rapidly forced potential flows with free boundaries. Determination of time-averaged stationary states leads to an optimal shape design problem. Under periodic boundary conditions existence and uniqueness of smooth minimizers to the averaged energy is proved using local coerciveness. In the numerical part of the article, 2D and 3D finite element approximations including related error estimates are discussed. Some illustrating examples are sketched.

We develop a functional-discrete method with a high order of accuracy to find a numerical solution of an eigenvalue transmission problem. It allows to approximate the trial eigenvalue with any desired accuracy. This approach has no restriction on the number of eigenvalues, an approximation to which can be found. The convergence rate is proved as in the case of the geometric series. It is shown that depending on the data of the original problem, two kinds of eigenvalue sequences may exist. For the first one, the convergence rate increases as the ordinal number of the trial eigenvalue increases. For the second one, the convergence rate is the same for all eigenvalues and does not depend on the ordinal number of the trial eigenvalue. Based on the asymptotic behavior of the eigenvalues of the basic problem and the functional-discrete method, a qualitative result on the arrangement of eigenvalues of the original problem is established. A number of numerical examples are given to support the theory.

Monotone economical difference schemes of the second order of local approximation with respect to space variables on nonuniform grids for the heat con- duction equation with the boundary conditions of the third kind in a p-dimensional parallelepiped are constructed. The a priori estimates of stability and convergence of the difference solution in the norm C are obtained by means of the grid maximum principle.

The discretization meshes of the Shishkin type are more suitable for high- order finite-difference schemes than Bakhvalov-type meshes. This point is illustrated by the construction of a hybrid scheme for a class of semilinear singularly perturbed reaction-diffusion problems. A sixth-order five-point equidistant scheme is used at most of the mesh points inside the boundary layers, whereas lower-order three-point schemes are used elsewhere. It is proved under certain conditions that this combined scheme is almost sixth-order accurate and that its error does not increase when the perturbation parameter tends to zero.