Band 2, Heft 2

In this paper, numerical algorithms for solving “fuzzy ordinary differential equations” are considered. A scheme based on the Taylor method of order p is discussed in detail and this is followed by a complete error analysis. The algorithm is illustrated by solving some linear and nonlinear fuzzy Cauchy problems.

The saturation assumption is widely used in a posteriori error analysis of finite element methods. It asserts, in its simplest form, that the solution can be approximated asymptotically with quadratic finite elements than with linear ones. In this paper, we show that a simple modification of this “hypothesis” is valid, and the proof of a posteriori error estimators by several authors becomes rigoruous with this simple modification.

In this paper, we present a finite difference analysis of the consolidation problem for saturated porous media. In the classical model, the behaviour of the porous environment – fluid system is described by a set of equations for the unknown vector displacements of the matrix skeleton and the fluid pressure. For simplicity we consider a model problem with constant coefficients in a rectangular domain. A priori estimates for the difference solution of the problem are obtained and on their basis the convergence of two-level difference schemes is investigated.

This paper is devoted to the construction of Newton type interpolation formulas for operators differentiable or not in the Gateaux sense based on interpolation by means of rational functions. The formulas obtained are exact for operator polynomials typical for the considered systems of rational functions.

For the abstract Cauchy problem for a parabolic equation a priori estimates of the global and asymptotic stability in various energy norms have been obtained. Similar problems are also considered for the second-order equation. In the latter case, a priori estimates of the asymptotic stability by the initial data have been obtained. The corresponding estimates of the global stability for three-level operator difference schemes have been proved. Estimates of the asymptotic behavior of the solution for quasi-linear multidimensional equations with unbounded nonlinearity have been obtained. The corresponding mathematical apparatus permitting one to prove unconditional monotonicity of the difference schemes approximating nonlinear problems is presented.

The main goal of this paper is to present a new strategy of increasing the convergence rate for the numerical solution of the linear finite element eigenvalue problems. This is done by introducing a postprocessing technique for eigenvalues. The postprocessing technique deals with solving a corresponding linear elliptic problem. We prove that the proposed algorithm has the superconvergence property of the eigenvalues and this improvement is attained at a small computational cost. Thus, good finite element approximations for eigenvalues are obtained on the coarse mesh. The numerical examples presented and discussed here show that the resulting postprocessing method is computationally more efficient than the method to which it is applied.

This is an attempt to construct a strong numerical method for transportdiffusion equations with nonlinear reaction terms, which relies on the idea of the Modified Method of Characteristics that is explicit but stable and is second-order accurate in time. The method consists in convective-diffusive splitting of the equations along the characteristics. The convective stage of the splitting is straightforwardly treated by a quasi-monotone and conservative modified method of characteristics, while the diffusive-reactive stage can be approximated by an explicit scheme with an extended real stability interval. A numerical comparative study of the new method with Characteristics Crank-Nicholson and Classical Characteristics Runge-Kutta schemes, which are used in many transport-diffusion models, is carried out for several benchmark problems, whose solutions represent relevant transport-diffusion-reaction features. Experiments for transport-diffusion equations with linear and nonlinear reactive sources demonstrate the ability of our new algorithm to better maintain the shape of the solution in the presence of shocks and discontinuities.