We study the approximation properties of the bivariate spline spaces S r 3r ( +) of smoothness r and degree 3r defined on triangulations + which are obtained from arbitrary nondegenerate convex quadrangulations by adding the diagonals of each quadrilateral.

. We derive L1 bounds for norms of projections onto bivariate polynomial spline spaces on regular triangulations with stable local bases. We then apply this result to derive error bounds for best L 2 - and ` 2 -approximation by splines on quasi-uniform triangulations. x1. Introduction Let X ` L1(\Omega\Gamma be a linear space defined a set\Omega with polygonal boundary. Suppose h\Delta; \Deltai is a semi-definite inner-product on X with associated semi-norm k \Delta k. We assume that hf; gi = 0, whenever fg = 0 on \Omega\Gamma (1:1) kfk kgk, whenever jf(x)j jg(x)j for all x 2\Omega : (1:2) Suppose S ` X is a linear space of polynomial splines (bivariate piecewise polynomials) defined on a regular triangulation 4 of\Omega (two triangles intersect only at a common vertex or along a common edge). We assume that S is a Hilbert space with respect to h\Delta; \Deltai. Let P : X ! S be the projection of X onto S defined by the minimization problem kf \Gamma Pfk = min s2S kf \Gamm...

. Stable locally supported bases are constructed for the spaces S r d (4) of polynomial splines of degree d 3r + 2 and smoothness r defined on triangulations 4, as well as for various superspline subspaces. In addition, we show that for r 1, it is impossible to construct bases which are simultaneously stable and locally linearly independent. x1. Introduction This paper deals with the classical space of polynomial splines S r d (4) := fs 2 C r(\Omega\Gamma : sj T 2 P d for all triangles T 2 4g; where P d is the space of polynomials of degree d, and 4 is a regular triangulation of a polygonal set\Omega\Gamma We also discuss superspline subspaces of the form S r;ae d (4) := fs 2 S r d (4) : s 2 C ae v (v) for all v 2 Vg; (1:1) with ae := fae v g v2V , where ae v are given integers such that r ae v d, and V is the set of all vertices of 4. Our aim is to describe algorithms for constructing locally supported bases fB i g 2I for these spaces which are stable in the follo...

Abstract. Macro-elements of arbitrary smoothness are constructed on Powell-Sabin triangle splits. These elements are useful for solving boundaryvalue problems and for interpolation of Hermite data. It is shown that they are optimal with respect to spline degree, and we believe they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over Powell-Sabin refinements. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power. 1.

We derive error bounds for bivariate spline interpolants which are calculated by minimizing certain natural energy norms. x1.

Bounds are provided on how well functions in Sobolev spaces on the sphere can be approximated by spherical splines, where a spherical spline of degree d is a C r function whose pieces are the restrictions of homogoneous polynomials of degree d to the sphere. The bounds are expressed in terms of appropriate seminorms defined with the help of radial projection, and are obtained using appropriate quasi-interpolation operators. x1.

Multivariate spline functions are smooth piecewise polynomial functions over triangulations consisting of n-simplices in the Euclidean space R^n. We present a straightforward method for using these spline functions to numerically solve elliptic partial differential equations such as Poisson and biharmonic equations and fit given scattered data. This method does not require constructing macro-elements or locally supported basis functions nor computing the dimension of the finite element spaces or spline spaces. We have implemented the method in MATLAB using multivariate splines in R² and R³. Several numerical examples are presented to demonstrate the effectiveness and efficiency of the method.

Macro-elements of arbitrary smoothness are constructed on Clough-Tocher triangle splits. These elements can be used for solving boundaryvalue problems or for interpolation of Hermite data, and are shown to be optimal with respect to spline degree. We conjecture they are also optimal with respect to the number of degrees of freedom. The construction provides local bases for certain superspline spaces defined over Clough-Tocher refinements of arbitrary triangulations. These bases are shown to be stable as a function of the smallest angle in the triangulation, which in turn implies that the associated spline spaces have optimal order approximation power. x1.

. Macro-elements of smoothness C r on Powell-Sabin triangle splits are constructed for all r 0. These new elements are improvements on elements constructed in [10] in that certain unneeded degrees of freedom have been removed. x1. Introduction A bivariate macro-element defined on a triangle T consists of a finite dimensional linear space S defined on T , and a set of linear functionals forming a basis for the dual of S. It is common to choose the space S to be a space of polynomials or a space of piecewise polynomials defined on some subtriangulation of T . The members of , called degrees of freedom, are usually taken to be point evaluations of derivatives. A macro-element defines a local interpolation scheme. In particular, if f is a sufficiently smooth function, then we can define the corresponding interpolant as the unique function s 2 S such that s = f for all 2 . We say that a macro-element has smoothness C r provided that if the element is used to construct an interpolati...

Let \Delta be a triangulation of some polygonal domain\Omega ae R 2 and let S r q (\Delta) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to \Delta. We present a Hermite type interpolation scheme for S r q (\Delta), q 3r + 2, that possesses optimal approximation order O(h q+1 ). Furthermore, the fundamental functions of our scheme form a locally linearly independent basis for a superspline subspace of S r q (\Delta). 1991 Mathematics Subject Classification. Primary 41A25, 41A63; Secondary 41A05, 41A15, 65D05, 65D07. Key words and phrases. Bivariate spline, triangulation, interpolation, approximation order, locally linearly independent basis. Research of the first author supported in part by a Research Fellowship from the Alexander von Humboldt Foundation. 1 Introduction Let\Omega ae R 2 be a polygonal domain, and let \Delta denote a regular triangulation of \Omega\Gamma The space of bivariate polynomial splines of degr...