We present the convergence analysis of an efficient numerical method for the solution of an initial-boundary value problem for a scalar nonlinear conservation law equation with a diffusion term. Nonlinear convective terms are approximated with the aid of a monotone finite volume scheme considered over the finite volume barycentric mesh, whereas the diffusion term is discretized by piecewise linear nonconforming triangular finite elements. Under the assumption that the triangulations are of weakly acute type, with the aid of the discrete maximum principle, a priori estimates and some compactness arguments based on the use of the Fourier transform with respect to time, the convergence of the approximate solutions to the exact solution is proved, provided the mesh size tends to zero.

A boundary value problem for the Laplace equation with Dirichlet and Neumann boundary conditions on an equilateral triangle is transformed to a problem of the same type on a rectangle. This enables us to use, e.g., the cyclic reduction method for computing the numerical solution of the problem. By the same transformation, explicit formulae for all eigenvalues and all eigenfunctions of the corresponding operator are obtained.

For open sets with a piecewise smooth boundary it is shown that we can express a solution of the Robin problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series.

Phase-field systems as mathematical models for phase transitions have drawn a considerable attention in recent years. However, while they are suitable for capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis effects occurring during phase transition processes. To overcome this shortcoming of existing phase-field theories, the authors have recently proposed a new approach to phase-field models which is based on the mathematical theory of hysteresis operators developed in the past fifteen years. Well-posedness and thermodynamic consistency were proved for a phase-field system with hysteresis which is closely related to the model advanced by Caginalp in a series of papers. In this note the more difficult case of a phase-field system of Penrose-Fife type with hysteresis is investigated. Under slightly more restrictive assumptions than in the Caginalp case it is shown that the system is well-posed and thermodynamically consistent.

In this paper, we develop a thermodynamically consistent description of the uniaxial behavior of thermovisco-elastoplastic materials for which the total stress σ contains, in addition to elastic, viscous and thermic contributions, a plastic component σ p of the form σp(x,t)=Ρ[ε, θ(x,t)](x,t). Here ∈ and θ are the fields of strain and absolute temperature, respectively, and {Ρ[·, θ]}θ>0 denotes a family of (rate-independent) hysteresis operators of Prandtl-Ishlinskii type, parametrized by the absolute temperature. The system of momentum and energy balance equations governing the space-time evolution of the material forms a system of two highly nonlinearly coupled partial differential equations involving partial derivatives of hysteretic nonlinearities at different places. It is shown that an initial-boundary value problem for this system admits a unique global strong solution which depends continuously on the data.

Nonsensitiveness regions for estimators of linear functions, for confidence ellipsoids, for the level of a test of a linear hypothesis on parameters and for the value of the power function are investigated in a linear model with variance components.The influence of the design of an experiment on the nonsensitiveness regions mentioned is numerically demonstrated and discussed on an example.

We apply the method of reliable solutions to the bending problem for an elasto-plastic beam, considering the yield function of the von Mises type with uncertain coefficients. The compatibility method is used to find the moments and shear forces. Then we solve a maximization problem for these quantities with respect to the uncertain input data.

In this paper, our attention is concentrated on the GMRES method for the solution of the system (I−T)x=b of linear algebraic equations with a nonsymmetric matrix. We perform m pre-iterations y l+1 =T yl +b before starting GMRES and put y m for the initial approximation in GMRES. We derive an upper estimate for the norm of the error vector in dependence on the mth powers of eigenvalues of the matrix T Further we study under what eigenvalues lay-out this upper estimate is the best one. The estimate shows and numerical experiments verify that it is advisable to perform pre-iterations before starting GMRES as they require fewer arithmetic operations than GMRES. Towards the end of the paper we present a numerical experiment for a system obtained by the finite difference approximation of convection-diffusion equations.

On a closed convex set Z in ℝN with sufficiently smooth (W 2,∞) boundary, the stop operator is locally Lipschitz continuous from W 1,1([0,T]ℝN) × Z into W 1,1([0,T],ℝN). The smoothness of the boundary is essential: A counterexample shows that C 1-smoothness is not sufficient.

In 1995, Wahbin presented a method for superconvergence analysis called “Interior symmetric method,” and declared that it is universal. In this paper, we carefully examine two superconvergence techniques used by mathematicians both in China and in America. We conclude that they are essentially different.

The main goal of the paper is to give a variational formulation of the behaviour of bolt systems in rock mass. The problem arises in geomechanics where bolt systems are applied to reinforce underground openings by inserting steel bars or cables. After giving a variational formulation, we prove the existence and uniqueness and some other properties.

For contractive interval functions [g] we show that $$[g]([x]_\varepsilon ^{k_0 } ) \subseteq \operatorname{int} ([x]_\varepsilon ^{k_0 } )$$ results from the iterative process $$[x]^{k + 1} : = [g]([x]_\varepsilon ^k )$$ after finitely many iterations if one uses the epsilon-inflated vector $$[x]_\varepsilon ^k$$ as input for [g] instead of the original output vector [x] k . Applying Brouwer's fixed point theorem, zeros of various mathematical problems can be verified in this way.

In [Sv1] a new micromechanical approach to the prediction of creep flow in composites with perfect matrix/particle interfaces, based on the nonlinear Maxwell viscoelastic model, taking into account a finite number of discrete slip systems in the matrix, has been suggested; high-temperature creep in such composites is conditioned by the dynamic recovery of the dislocation structure due to slip/climb motion of dislocations along the matrix/particle interfaces. In this article the proper formulation of the system of PDE's generated by this model is presented, some existence results are obtained and the convergence of Rothe sequences, applied in the specialized software CDS, is studied.

Lanczos' method for solving the system of linear algebraic equations Ax=b consists in constructing a sequence of vectors x k in such a way that $$r_k = b - Ax_k \in \;\;r_0 + A\mathcal{K}_k (A,r_0 )$$ and $$r_k \bot \mathcal{K}_k (A^T ,\tilde r_0 )$$ . This sequence of vectors can be computed by the BiCG (BiOMin) algorithm. In this paper is shown how to obtain the recurrences of BiCG (BiOMin) directly from this conditions.

Let e t=(e t1,...e tp)′ be a p-dimensional nonnegative strict white noise with finite second moments. Let h ij(x) be nondecreasing functions from [0,∞) onto [0,∞) such that h ij(x) ≤ x for i, j = 1,...,p. Let U = (u ij) be a p×p matrix with nonnegative elements having all its roots inside the unit circle. Define a process X t=(X t1,...,X tp)′ by for $$X_{tj} = u_{j1} h_{1j} (X_{t - 1,1} ) + ... + u_{jp} h_{pj} (X_{t - 1,p} ) + e_{tj}$$ for j=1,..., p A method for estimating U from a realization X 1,...,X n is proposed. It is proved that the estimators are strongly consistent.

The linearized vorticity equation serves to model a number of wave phenomena in geophysical fluid dynamics. One technique that has been applied to this equation is the geometrical optics, or multi-dimensional WKB technique. Near caustics, this technique does not apply. A related technique that does apply near caustics is the Lagrange Manifold Formalism. Here we apply the Lagrange Manifold Formalism to determine an asymptotic solution of the linearized vorticity equation and to study associated wave phenomena on the caustic curve.

The concept of global statistical information in the classical statistical experiment with independent exponentially distributed samples is investigated. Explicit formulas are evaluated for common exponential families. It is shown that the generalized likelihood ratio test procedure of model selection can be replaced by a generalized information procedure. Simulations in a classical regression model are used to compare this procedure with that based on the Akaike criterion.

The paper gives the answer to the question of the number and qualitative character of stationary points of an autonomous detailed balanced kinetical system.

The aim of this paper is to characterize the Multivariate Gauss-Markoff model (MGM) as in (2.1) with singular covariance matrix and missing values. MGMDP2 model and completed MGMDP2Q model are obtained by three transformations D, P and Q (cf. (3.21)) of MGM. The unified theory of estimation (Rao, 1973) which is of interest with respect to MGM has been used.The characterization is reached by estimation of parameters: scalar σ2 and linear combination $$\lambda '\bar B\left( {\bar B = vecB} \right)$$ as in (4.8), (4.6), (4.7) as well as by the model of the form (5.1) (cf. Th. 5.1). Moreover, testing linear hypothesis in the available model MGMDP2 by test function F as in (6.3) and (6.4) is considered.It is known (Oktaba 1992) that ten quantities in models MGMDP2, and MGMDP2Q are identical (invariant). They permit to say that formulas for estimation and testing in both models are identical (Oktaba et al., 1988, Baksalary and Kala, 1981, Drygas, 1983).An algorithm and the UMGMBO program for calculations concerning estimation and testing in MGM have been presented by Oktaba and Osypiuk (1993).

In this paper, a method of numerical solution to the dominant eigenvalue problem for positive integral operators is presented. This method is based on results of the theory of positive operators developed by Krein and Rutman. The problem is solved by Monte Carlo method constructing random variables in such a way that differences between results obtained and the exact ones would be arbitrarily small. Some numerical results are shown.