Volume 18, Issue 3

In this paper we consider a variant of the additive Schwarz preconditioner for elliptic finite element problems on meshes which are strongly refined in the vicinity of the Neumann boundary condition. Local refinements can be motivated by a sharp boundary layer in the solution of the underlying differential equation or they may occur in a method for solving another differential problem. The classical example of the latter situation is the L 2 -projection (or the splitting-up) method for the Navier–Stokes equations. For a model problem we introduce an additive Schwarz preconditioner and prove that the condition number of the preconditioned stiffness matrix is bounded from above by a constant which is independent of the mesh and the width of the boundary layer. We consider several generalizations and applications of the results to the model problem. In particular, we discuss the application of the proposed preconditioner in the mortar element method. Numerical computations confirm the theoretical results of the paper.

We consider the method of reconstructing the concentration of species and gases in the atmosphere by the measured values of intensity of radiation emitted and absorbed in various radiation frequency bands. At the first step, we reconstruct the altitude profile of the total absorption factor for different observation frequencies. At the second step, we reconstruct the components concentrations at different altitudes. The inverse problem is ill-posed, statistical regularization is used for its solution. The reconstruction method proposed is compared to the standard one.

In this paper, we consider problems involving the construction of adjoint equations for nonlinear equations of mathematical physics. Hydrodynamical-type systems, in particular, dynamic equations for two-dimensional incompressible ideal fluid are taken as the main subject of investigation. It is shown that using adjoint equations, not only can we construct the known integrals of motion, but also obtain new integrals that are useful, in particular, for investigating the stability of solutions of the original equations. It is also shown that the nonuniqueness of the construction of adjoint equations for original nonlinear problems can be used to construct the finite-dimensional approximations of the original equations. These approximations have the necessary set of finite-dimensional analogues of integral conservation laws. The algorithm for constructing these schemes is given for a problem of two-dimensional ideal incompressible fluid dynamics.

We consider a barotropic inviscid model of chaotic advection in a unidirectional pulsating background flow over a seamount of an elliptic form. We numerically study the process of passive tracer transport from the vortex area to the flow-through area, and in particular we give the dependence of evolution of the corresponding Poincaré maps on the frequency and amplitude of incident flow pulsations. We propose an approach to the study of the mechanism and parameters of chaotic advection in open systems with finite trajectories lifetime. It is based on studying the time it takes for tracers to be carried out from the vortex area to the flow-through area. We show the essential impact of the seamount orientation with respect to an incident flow on the rate and scenario of tracers transport from the vortex area.

In the framework of mixed and hybrid finite element methods for diffusion-type and other elliptic differential equations, it is necessary to apply numerical schemes with variables given as values of normal fluxes on the edges (faces) of the elementary cells and values of the scalar-valued function in each cell. In this paper, we propose a general method of constructing finite element approximations on polygonal and polyhedral meshes, whose cells are convex and nonconvex polygonal domains in and polyhedrons in . We present a natural way of constructing cell prolongation operators, which makes it possible to easily compute the coefficients of the respective mass matrices. Also, the proposed prolongations satisfy the important requirement that the image of the divergence operator on the extended fields belongs to the set of piecewise constant functions. The latter fact provides direct justification of the well–posedness of the arising discrete problems.