Volume 17, Issue 6

- In this paper we introduce and investigate a variant of the additive Schwarz preconditioner for elliptic finite element problems on meshes with strong refinement in the vicinity of the Dirichlet part of the boundary. This type of local refinement can be due to either a sharp boundary layer in the solution of the differential equation or singularities in its derivatives. The computational domain is decomposed into a large size subdomain containing a major part of the mesh and a number of small size subdomains containing the refined part of the mesh. The additive Schwarz method based on this sort of domain decomposition is proved to be optimal in the sense that the condition number of the preconditioned stiffness matrix is bounded from above by a constant which is independent of the mesh and the sizes of small subdomains.

- We numerically investigate the convergence rate behaviour of various iterative methods of solving the bilinear and bicubic finite element approximations of boundary value problems for the dissipative singularly perturbed Helmholtz equation with large values of the singular parameter μ 2 . As μĥ increases, where h is a reduced grid step, the efficiency of the multigrid method decreases and the conventional overrelaxation method for μĥ ≥ 1 turns out to be faster. The alternating direction method with the optnnal set of parameters turns out to be the fastest one for large values of μĥ (for domams which admit the separation of variables). We obtain a simple efficient approximate formula for calculating these parameters, derive its error estimates. The investigation of the spectra bounds of one-dnnensional grid operators in the case of a bicubic FE scheme for the Neumann problem results in the introduction of a more correct modification of this scheme, which is of higher-order accuracy for the derivatives of the solutions.

To solve the nondegenerate system of linear special-type equations we consider a threeparameter algorithm which is a generalization of the known Arrow-Hurwicz algorithm. For the proposed algorithm we solve an asymptotic optimization problem in two different statements. We obtain analytic formulae for asymptotically optimal parameters and the convergence factor. The results obtained are of an unimprovable character for the studied class of algorithms.

- In this work we construct new algorithms of the method of statistical simulation for the numerical solution of the third boundary value problem for a parabolic-type equation. They are based on a combination of the probabilistic representation of the solution and the difference approximation of the boundary condition. We derive the estimate of the solution to the boundary value problem using the diffusion process with a rebound from the boundary. Unlike the available schemes for the weak approximation, in the approach proposed we do not use the local time of the diffusion process, which substantially complicates numerical estimates

- We present a new method of ocean dynamics modelling, taking into account the gridscale turbulent advection (GSTA). By GSTA we mean the average values of the products of prognostic variables pulsation (the second moments). They are taken into account by the method of successive approximations. It is shown that the role of the effect of gid-scale (EGS) turbulence is rather important: this effect prevents the drastic fall of KE (kinetic energy) in quasi-prognostic calculations, keeping it on a level up to 3-5 times as large as the direct prognostic value of KE. These preliminary calculations are done at the initial stage of adjustment.`