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Open Access
January 1, 2001
### Abstract

We consider an L_2-norm least-squares principle for a scalar hyperbolic problem. A proper variational framework for the associated finite element method is developed and studied. Analysis of the discretization error based on the least-squares projection property shows a gap of one. This number cannot be improved with a standard duality argument because the least-squares dual does not possess full elliptic regularity. Using a perturbed dual problem we are able to show that the actual gap of the least-squares method in the constant convection case is not worse than 2/3.

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Open Access
January 1, 2001
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This work discusses issues on the design and analysis of finite difference schemes for 3D modeling the process of moisture motion in the wood. A new finite difference scheme is proposed. The stability and convergence in the maximum norm are proved for Robin boundary conditions. The influence of boundary conditions is investigated, and results of numerical experiments are presented.

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Open Access
January 1, 2001
### Abstract

The problem of approximating the band structure of electromagnetic Bloch modes in a three-dimensional periodic medium is studied. We analyze a mixed finite element approximation technique based on a variation of Nedelec edge elements. The usual conditions for convergence of the static problem are first verified. Subsequently, convergence of approximate eigenvalues to those of the continuous system is proved.

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Open Access
January 1, 2001
### Abstract

Singularly perturbed quasilinear boundary value problems exhibiting boundary layers are considered. Special piecewise-uniform meshes are constructed which are fitted to these boundary layers. Numerical methods composed of upwind difference operators and these fitted meshes are shown to be parameter robust, in the sense that the solutions satisfy an error estimate in the maximum norm which is independent of the value of the singular perturbation parameter. Numerical results supporting the theory are presented.

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Open Access
January 1, 2001
### Abstract

In the present work, a symmetrized sequential-parallel decomposition method of the third degree of precision for solving Cauchy abstract problem is suggested. The third degree of precision is reached by introducing the complex coefficient α=1/2±i/(2√3)s. For the scheme considered, explicit a priori estimation is obtained.

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Open Access
January 1, 2001
### Abstract

In this paper we consider the numerical approximation of the solution of the 2D unsteady Lame equations on a rectangular domain. The basic problems that appear, using both finite difference and finite element methods, are connected with the fact that these equations are strongly coupled. Thus it is natural to design computational algorithms in such a way that they allow one to consider boundary value problems only for uncoupled equations. To implement this general concept, some special (unconditionally stable) operator-splitting schemes are constructed. Its major peculiarity is that transition to the next time level is performed by solving separate elliptic problems for each component of the displacement vector. The previous results make it possible to design efficient numerical algorithms for elasticity equations.

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Open Access
January 1, 2001
### Abstract

A computational scheme for approximating a singular operator in the known Lippman-Schwinger equations is suggested. It is based on partial use of Gaussian knots. On the basis of studying the asymptotic properties of the solution and one unclassical estimation of the accuracy of the Gaussian quadrature formulas, the convergence of the scheme, with estimation of the rate of convergence, is proved. The analysis of the computational aspects of the scheme is carried out and the results of calculations of some test examples are given.