Volume 20, Issue 4

Non-conforming finite element approximation of the shallow water equations in the spherical coordinate system is constructed. Peculiarities of computational realization are discussed.

This paper deals with the description of kinetic schemes and the quasi-gas-dynamic system of equations, which is a promising trend for modelling complex gas dynamic flows. The possibilities of kinetic schemes and the quasi-gas-dynamic system as a model for describing a viscous gas flow are discussed. Particular attention is given to the possibility of their application to the computation of gas dynamics problems on high-performance multiprocessor computer systems. The possibilities to use the specified approaches for the solution of other problems of continuum mechanics are considered.

When numerically implementing the optimal binomial N -cycle Chebyshev iterative methods for solving the linear operator equations Au = f [1–13], the iterative sequence of approximations u k , k = 1, . . . , N , arises. In this case a transition operator is successively multiplied by operator factors of the form I − α k A , k ≤ N , where the parameters α k are inverse values to the roots of the Chebyshev first-kind polynomial of the degree N reduced to the segment [ m , M ], 0 < m < M , which comprises the spectrum of the operator A . For small values of ξ = m / M the norms of some of these operators are large. It is advisable to mix these operators with those which reduce the norm of the transition operator. An important problem is the stability of the iterative method u k +1 = u k – α k ( Au k – f ), k = 0,1, . . . , N –1. Its stability to round-off errors substantially depends on the sequence of parameters (permutation ϰ N ), because the transition operators I – α k Aξ for small values of ξ = m / M and some numbers k have a huge norm. In real computer-aided calculations this can lead to two undesirable effects: (1) a dramatic increase in || u k || for some 1 ≤ k < N, which can result in an emergency halt of the computer or a loss in significant digits in subsequent calculations, and (2) the corresponding increase in the errors at intermediate iterations. The algorithms for ordering parameters for the case N = 2 q 3 r have been studied in [1–6, 8, 9].

This paper concerns a posteriori error estimation for the primal and dual mixed finite element methods applied to the diffusion problem. The problem is considered in a general setting with inhomogeneous mixed Dirichlet–Neumann boundary conditions. New functional-type a posteriori error estimators are proposed that exhibit the ability both to indicate the local error distribution and to ensure upper bounds for discretization errors in primal and dual (flux) variables. The latter property is a direct consequence of the absence in the estimators of any mesh-dependent constants; the only constants present in the estimates stem from the Friedrichs and trace inequalities and, thus, are global and dependent solely on the domain geometry and the bounds of the diffusion matrix. The estimators are computationally cheap and require only the projections of piecewise constant functions onto the spaces of the lowest-order Raviart–Thomas or continuous piecewise linear elements. It is shown how these projections can be easily realized by simple local averaging.

We proposed a fast algorithm for computing certain multilevel Toeplitz forms. It helps to implement some simple quadrature formulae in an efficient way allowing for the use of more nodes. An application example is given with volume integral equations in electromagnetics.

Black-box methodology for generating anisotropic adaptive tetrahedral meshes in domains with discrete boundaries is described. A new high-order reconstruction method for triangular surface meshes is proposed. The performance of the method for a model convection–diffusion problem is demonstrated.