Numerical integration based on bivariate quadratic spline quasi-interpolants on Powell-Sabin partitions

In this paper we generate and study new cubature formulas based on spline quasi-interpolants in the space of quadratic Powell-Sabin splines on nonuniform triangulations of a polygonal domain in ?². By using a specific refinement of a generic triangulation, optimal convergence orders are obtained for some of these rules. Numerical tests are presented for illustrating the theoretical results.     

Existence and Construction of H 1-Splines of Class C k on a Three Directional Mesh

Let be the uniform triangulation generated by the usual three directional mesh of the plane and let H ₁ be the regular hexagon formed by the six triangles of surrounding the origin. We study the space of piecewise polynomial functions in C ^k (R ²) with support H ₁ having a sufficiently high degree n, which are invariant with respect to the group of symmetries of H ₁ and whose sum of integer translates is constant. Such splines are called H ₁-splines. We first compute the dimension of this space in function of n and k. Then we prove the existence of a unique H ₁-spline of minimal degree for any fixed k0. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of this spline.     

A collocation method for the numerical solution of a two dimensional integral equation using a quadratic spline quasi-interpolant

In this paper, we propose an interesting method for approximating the solution of a two dimensional second kind equation with a smooth kernel using a bivariate quadratic spline quasi-interpolant (abbr. QI) defined on a uniform criss-cross triangulation of a bounded rectangle. We study the approximation errors of this method together with its Sloan’s iterated version and we illustrate the theoretical results by some numerical examples.     

Normalized trivariate B-splines on Worsey-Piper split and quasi-interpolants

In this work, we give an algorithm for constructing a normalized B-spline basis over a Worsey-Piper split of a bounded domain of ℝ³. These B-splines are all positive, have local support and form a partition of unity. Therefore, they can be used for constructing local approximants and for many other applications in CAGD. We also introduce the Worsey-Piper B-spline representation of C ¹ quadratic polynomials or splines in terms of their polar forms. Then, we use this B-representation for constructing several quasi-interpolants which have an optimal approximation order.     

A General Multiresolution Method for Fitting Functions on the Sphere

In [7], Lyche and Schumaker have described a method for fitting functions of class C ¹ on the sphere which is based on tensor products of quadratic polynomial splines and trigonometric splines of order three associated with uniform knots. In this paper, we present a multiresolution method leading to C ²-functions on the sphere, using tensor products of polynomial and trigonometric splines of odd order with arbitrary simple knot sequences. We determine the decomposition and reconstruction matrices corresponding to the polynomial and trigonometric spline spaces. We describe the general tensor product decomposition and reconstruction algorithms in matrix form which are convenient for the compression of surfaces. We give the different steps of the computer implementation of these algorithms and, finally, we present a test example.     

C 1 Quadratic and C 2 Quartic Macro-Elements on a Modified Morgan-Scott Triangulation

In this paper, we construct a quadratic composite finite element of class C ¹ and quartic composite finite element of class C ² on a new triangulation τ ₁₀ which is obtained by splitting each triangle of a given triangulation τ into ten smaller subtriangles. These new elements can be used for constructing spline spaces with local basis that can be applied for solving some Hermite interpolation problems with optimal approximation order.     

Collocation methods for solving multivariable integral equations of the second kind

In a recent paper (Allouch, in press) [5] on one dimensional integral equations of the second kind, we have introduced new collocation methods. These methods are based on an interpolatory projection at Gauss points onto a space of discontinuous piecewise polynomials of degree r$r$ which are inspired by Kulkarni’s methods (Kulkarni, 2003) [10], and have been shown to give a 4r+4$4r+4$ convergence for suitable smooth kernels. In this paper, these methods are extended to multi-dimensional second kind equations and are shown to have a convergence of order 2r+4$2r+4$. The size of the systems of equations that must be solved in implementing these methods remains the same as for Kulkarni’s methods. A two-grid iteration convergent method for solving the system of equations based on these new methods is also defined.     

A multiresolution method for fitting scattered data on the sphere

In this work, we propose an efficient multiresolution method for fitting scattered data functions on a sphere S, using a tensor product method of periodic algebraic trigonometric splines of order 3 and quadratic polynomial splines defined on a rectangular map of S. We describe the decomposition and reconstruction algorithms corresponding to the polynomial and periodic algebraic trigonometric wavelets. As application of this method, we give an algorithm which allows to compress scattered data on spherelike surfaces. In order to illustrate our results, some numerical examples are presented.