Katarina Surla

Solving singularly perturbed boundary-value problems by spline in tension

The difference scheme via spline in tension for the problem: − ϵy″ + p(x)y = f(x), p(x)\s>0, y(0) = α0, y(1) = α1, is derived. The error of the form O(h min(h, √ϵ) is obtained. When p(x) = p = const., the corresponding spline in tension achieves the second order of the global uniform convergence.

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.

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.

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