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A parallel version of a multigrid method [Z. Chen, On the convergence of Galerkin-multigrid methods for nonconforming finite elements. East-West J. Numer. Math . (1999) 7, No. 2, 79–104.] for a face-oriented nonconforming finite element space is considered. Primary attention is focused on a data structure providing simple implementation. In particular, a technique of parallelization is derived in a logically consistent way from a set of a priori formulated requirements to the parallel implementation.

In this paper we find and study the class of symmetric methods among the Runge-Kutta formulae. It is shown that the explicit Runge-Kutta methods cannot be symmetric. We also define the conditions which coefficients in the implicit Runge-Kutta method should satisfy for it to be symmetric. Particular attention has been given to the study of stability properties in the symmetric Runge-Kutta formulae. It is proved that in some cases the notions of absolute and algebraic stability for the given class of numerical methods coincide. Besides, we find a restriction to the order of stable symmetric methods among the diagonally implicit Runge-Kutta formulae. Finally, we give full characteristics of all algebraically stable symmetric Runge-Kutta methods in terms of a transformed matrix of coefficients.

In this paper, we consider circulation anomalies obtained in the model of atmospheric general circulation when incoming short wave radiation changes and the initial state anomalies are given. We also consider circulation differences during the years of high and low moisture content in the atmosphere, using the reanalysis data. Using the global model, we studied such winter and summer circulation anomalies that result from the attenuation of winter radiation forcing and accordingly reduction in summer radiation atmosphere heating. In the other series of experiments the joint 'atmosphere-soil' initial moisture anomalies in transitional seasons (autumn, spring) are formed by using a hemisphere model and their effects on the subsequent structure of the circulation anomalies are investigated. It turned out that these anomalous numerical experiments different both in the boundary and initial conditions model the similar anomalies. Namely, in transition in the annual variation to winter circulation (or in winter) pressure long-wave troughs and ridges are damped, in transition to summer circulation (or in summer) west-east transport at the high latitudes is attenuated by the modelled anomalies.

This paper deals with the derivation of two-sided a posteriori error estimates for approximate solutions of boundary value problems in the linear theory of elasticity. It is shown that the estimates are consistent, explicitly computable, and valid for any approximation of the respective energy space. Numerical tests are conducted for plane strain problems in domains with reentrant corners. They confirm practical efficiency of these estimates.