Zorica Uzelac

The Sdfem for a Convection-diffusion Problem with Two Small Parameters

Abstract A singularly perturbed convection-diffusion problem with two small parameters is considered. The problem is solved using the streamline-diffusion finite element method on a Shishkin mesh. We prove that the method is convergent independently of the perturbation parameters. Numerical experiments support these theoretical results.

Researching indicators of organizational intellectual capital in Serbia

Purpose – The purpose of this paper is to provide an appropriate model for IC reporting in the transitional economic system of Serbia.Design/methodology/approach – The existing methods of IC reporting as well as the actual needs of organizations in Serbia related to measuring IC (i.e. key influencing factors of IC) and their unique features represent the fundamentals of an adequate model for IC reporting.Findings – A group of selected IC indicators in Serbia differs from typical relevant indicators mainly due to the specifics of the environment. Preliminary investigation of organizational IC in Serbia was carried out and a number of initiatives for improvement and development of the existing IC have been suggested.Research limitations/implications – An implementation of the proposed model is limited to the observed environment. The suggested group of indicators should be viewed more as a basis for a general application of IC reporting and management in Serbia, rather than an absolutely accurate model for ...

A numerical-asymptotic solution for singular perturbation problems

A class of singular perturbation problems is considered. In order to solve them a numerical-asymptotic method is constructed in which asymptotic solution techniques are combined with standard numerical methods. The proposed method is distinguished by the following facts: First, we construct the division point which divides the initial interval into two subintervals, so that the layer belongs only to one of them. The division point is constructed in such a way that it provides the best possible accuracy of such a combined method. The inner solution problem is solved as a two point boundary value problem, where the terminal boundary conditions are supplied by the solution of reduced problem. As the outer solution the solution of reduced problem is taken. A numerical example is included.

The discrete minimum principle for quadratic spline discretization of a singularly perturbed problem

We consider a singularly perturbed boundary value problem with two small parameters. The problem is numerically treated by a quadratic spline collocation method. The suitable choice of collocation points provides the discrete minimum principle. Error bounds for the numerical approximations are established. Numerical results give justification of the parameter-uniform convergence of the numerical approximations.

Family of quadratic spline difference schemes for a convection-diffusion problem

We consider the finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. It is discretized using quadratic splines as approximation functions, equations with various piecewise constant coefficients as collocation equations and a piecewise uniform mesh of Shishkin type. The family of schemes is derived using the collocation method. The numerical methods developed here are non-monotone and therefore apart from the consistency error we use Greens grid function analysis to prove uniform convergence. We prove the almost first order of convergence and furthermore show that some of the schemes have almost second-order convergence. Numerical experiments presented in the paper confirm our theoretical results.

A robust layer-resolving spline collocation method for a convection–diffusion problem

Abstract We consider finite difference approximation of a singularly perturbed one-dimensional convection–diffusion two-point boundary value problem. The problem is numerically treated by a quadratic spline collocation method on a piecewise uniform slightly modified Shishkin mesh. The position of collocation points is chosen so that the obtained scheme satisfies the discrete minimum principle. We prove pointwise convergence of order O ( N - 2 ln 2 N ) inside the boundary layer and second order convergence elsewhere. The uniform convergence of the approximate continual solution is also given. Further, we approximate normalized flux and give estimates of the error at the mesh points and between them. The numerical experiments presented in the paper confirm our theoretical results.

An optimal uniformly convergent oci difference scheme for a singular perturbation problem

The exponentially fitted tridiagonal OCI difference scheme for singularly perturbed problem is derived. The error at the grid points is bounded by , is a constant independent of ∊ and h, h is the step of a uniform mesh. The scheme has the maximal order of the classical and the uniform accuracy.

A modified Bakhvalov mesh

Abstract We consider a few numerical methods for solving a one-dimensional convection–diffusion singularly perturbed problem. More precisely, we introduce a modified Bakvalov mesh generated using some implicitly defined functions. Properties of this mesh and convergence results for several methods on it are given. Numerical results are presented in support of the theoretical considerations.

Qualocation for a singularly perturbed boundary value problem

A singularly perturbed one-dimensional two point boundary value problem of reaction-convection-diffusion type is considered. We generate a C^0-collocation-like method by combining Galerkin with an adapted quadrature rule. Using Lobatto quadrature and splines of degree r, we prove on a Shishkin mesh for the qualocation method the same error estimate as for the Galerkin technique. The result is also important for the practical realization of finite element methods on Shishkin meshes using quadrature formulas. We report the results of numerical experiments that support the theoretical findings.

An analysis of a uniformly accurate spline difference method

We are concerned with the numerical approximation by finite difference techniques of the linear two-point boundary value problem and d(x)≥0. Using exponential splines from C 1[0, 1] as approximation functions and equations with various piecewise constant coefficients as the collocation equations a class of uniformly second order accurate schemes is derived. It is proved for the scheme called IIEMW scheme that the errors at the grid points are bounded by Mh 2 whend(x)=0, Mis a constant independent of ∊ and step size h. The numerical results show that the estimate is valid when d(x)≠0. The IIEMW scheme, the well known EI-Mistikawy and Werle (the EMW scheme) and another one, called the IEMW scheme, from the family having second order accuracy, are analysed and compared