Basis of splines associated with singularly perturbed advection -diffusion problems

Among fitted-operator methods for solving one-dimensional singular perturbation problems one of the most accurate is the collocation by linear combinations of ${1,x,exp{(pm p x)} }$, known as tension spline collocation. There exist well established results for determining the `tension parameter\u27 $p$, as well as special collocation points, that provide higher order local and global convergence rates. However, if the advection-diffusion reaction problem is specified in such a way that two boundary internal layers exist, the method is incapable of capturing only one boundary layer, which happens when no reaction term is present. For a pure advection-diffusion problem we therefore modify the basis accordingly, including only one exponential, i.e. project the solution to the space locally spanned by ${1,x,x^2,exp{(p x)}}$ where $p>0$ is the tension parameter. The aim of the paper is to show that in this situation it is still possible to construct a basis of $C^1$-locally supported functions by a simple knot insertion technique, commonly used in computer aided geometric design. We end by showing that special collocation points can be found, which yield better local and global convergence rates, similar to the tension spline case

Constrained modification of the cubic trigonometric Bézier curve with two shape parameters

A new type of cubic trigonometric Bézier curve has been introduced in [1]. This trigonometric curve has two global shape parameters λ and µ. We give a lower boundary to the shape parameters where the curve has lost the variation diminishing property. In this paper the relationship of the two shape parameters and their geometric effect on the curve is discussed. These shape parameters are independent and we prove that their geometric effect on the curve is linear. Because of the independence constrained modification is not unequivocal and it raises a number of problems which are also studied. These issues are generalized for surfaces with four shape parameters. We show that the geometric effect of the shape parameters on the surface is parabolic. Keywords: trigonometric curve, spline curve, constrained modificatio

Geometric properties and constrained modification of trigonometric spline curves of Han

New types of quadratic and cubic trigonometrial polynomial curves have been introduced in [2] and [3]. These trigonometric curves have a global shape parameter λ. In this paper the geometric effect of this shape parameter on the curves is discussed. We prove that this effect is linear. Moreover we show that the quadratic curve can interpolate the control points at λ = √2. Constrained modification of these curves is also studied. A curve passing through a given point is computed by an algorithm which includes numerical computations. These issues are generalized for surfaces with two shape parameters. We show that a point of the surface can move along a hyperbolic paraboloid

Piecewise Extended Chebyshev Spaces: a numerical test for design

Given a number of Extended Chebyshev (EC) spaces on adjacent intervals, all of the same dimension, we join them via convenient connection matrices without increasing the dimension. The global space is called a Piecewise Extended Chebyshev (PEC) Space. In such a space one can count the total number of zeroes of any non-zero element, exactly as in each EC-section-space. When this number is bounded above in the global space the same way as in its section-spaces, we say that it is an Extended Chebyshev Piecewise (ECP) space. A thorough study of ECP-spaces has been developed in the last two decades in relation to blossoms, with a view to design. In particular, extending a classical procedure for EC-spaces, ECP-spaces were recently proved to all be obtained by means of piecewise generalised derivatives. This yields an interesting constructive characterisation of ECP-spaces. Unfortunately, except for low dimensions and for very few adjacent intervals, this characterisation proved to be rather difficult to handle in practice. To try to overcome this difficulty, in the present article we show how to reinterpret the constructive characterisation as a theoretical procedure to determine whether or not a given PEC-space is an ECP-space. This procedure is then translated into a numerical test, whose usefulness is illustrated by relevant examples