Volume 16, Issue 3

The properties of algebraic systems for the serendipity FEM with the hierarchical basis functions are investigated for an anisotropic two-dimensional boundary value problem. The constants of the spectral equivalence of the matrices are obtained. Several one- and two-level iterative processes are considered, including the defect-correction approach. Theoretical estimates are confirmed by the results of numerical experiments.

In this paper we propose a new approach to the construction of explicit schemes for solving boundary value parabolic problems. It is based on the block representation of a grid operator with substantially different spectral properties of blocks. It is shown that with the proper choice of integration steps the stability of explicit schemes is ensured by conditions independent for each of the blocks. We obtain stability estimates with respect to the initial data and the right-hand side. Unlike the earlier studies on the same subject, in this paper we discuss a multilevel approach with conditions of the Neumann type

In order to solve the system of nonlinear partial differential equations which describes the 3D ion and electron dynamics in the plasma wakefield we propose a difference scheme and an iteration algorithm for realizing it. The results of numerical modelling show that a low ion density region (the so-called plasma channel), which results in the breaking of the wakefield, forms behind a short laser pulse.

We consider the problem of data assimilation for a quasilinear parabolic equation to identify the initial condition. We study the properties of the nonlinear operator of the problem and prove the solvability of the data assimilation problem in a class of functional spaces. The Newton algorithm for the optimality system is given.

We study a new approach to the approximation of solutions of stochastic differential equations (SDEs) with reflection at a boundary for multidimensional domains with sufficiently smooth boundary. The approach is based on an arbitrary numerical method of solving SDEs with reflection from the hyperplane and on the Ito formula. Simultaneously with the approximation of the solution we calculate the estimate of the so-called local time of the process at the boundary. We give algorithms for calculating the estimates of solutions to the second and third boundary value problems, which are based on the probabilistic representations of the solutions and on the approach proposed. In conclusion we present a new method of calculating the first eigenvalue of an arbitrary elliptic operator in the class of functions that satisfy the homogeneous boundary Dirichlet condition or a homogeneous mixed condition. This method is based on the calculation of parametric derivatives.