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This paper is devoted to the numerical study for a class of boundary value problems of singularly perturbed linear second-order differential-difference equations with small shifts (i. e., containing both terms having a negative shift and terms having a positive shift). In particular, the numerical study for the problems where second order derivative is multiplied by a small parameter e and the shifts depend on the small parameter have been considered. To obtain a parameter-uniform convergence, a piecewise-uniform mesh is constructed, which is dense in the boundary layer region and coarse in the outer region. The parameter-uniform convergence analysis of the method has been discussed. The method is shown to have almost second order parameter-uniform convergence. The effect of small shifts on boundary layers have also been discussed. To demonstrate the efficiency of the proposed scheme several examples having boundary layers have been carried out.

This article continues prior work of the authors on the combined a posteriori analysis for the discretization and iteration errors in the finite element approximation of linear elliptic problems to the nonlinear case. The underlying theoretical framework is again that of the Dual Weighted Residual (DWR) method for goal-oriented error control. The accuracy in the algebraic solution process can be balanced with that due to discretization using computable a posteriori error estimates in which the outer nonlinear and inner linear iteration errors are separated from the discretization error. This results in effective stopping criteria for the algebraic iteration, which are elaborated particularly for Newton-type methods. The performance of the proposed strategies is demonstrated for several nonlinear test problems.

-The presented paper focuses on two objectives, the development of a new curve interpolation scheme and its application towards the visualization of shaped data. A C 1 piecewise rational cubic function, with two parameter family, is developed and presented. The affect of parameters on the shape of the model curve is examined mathematically and illustrated graphically. Simple sufficient data dependent constraints are obtained on one family of the parameters to visualize positive, monotone and convex data. However, the other family of parameters can assume any positive values. The problem of visualization of constrained data is also addressed when the data is lying above a straight line and curve is required to lie on the same side of the line. Moreover, the approximation order of the proposed rational cubic function is also investigated and is found to be O ( h 3 i ).