Klaus Höllig

High accuracy geometric Hermite interpolation

We describe a parametric cubic spline interpolation scheme for planar curves which is based on an idea of Sabin for the construction of C^1 bicubic parametric spline surfaces. The method is a natural generalization of [standard] Hermite interpolation. In addition to position and tangent, the curvature is prescribed at each knot. This ensures that the resulting interpolating piecewise cubic curve is twice continuously differentiable with respect to arclength and can be constructed locally. Moreover, under appropriate assumptions, the interpolant preserves convexity and is 6-th order accurate.

B-Splines From Parallelepipeds.

Abstract : Local support bases for piecewise polynomial spaces are important for applications such as finite element methods, data fitting etc. In (BH sub 1) a general construction principle for such B-splines was described. A special case are the so called box-splines. They have a particularly regular discontinuity pattern and coincide in special cases with standard finite elements. It is hoped that using translates of box-splines will lead, at least in two variables, to a unified theory for piecewise polynomial functions on regular meshes. This note is a first attempt in this direction and deals with basic approximation properties of translates of one box-splines such as stability, degree of approximation etc. (Author)

Weighted Extended B-Spline Approximation of Dirichlet Problems

We describe a new finite element method which uses weighted extended B-splines on a regular grid as basis functions for solving Dirichlet problems on bounded domains in arbitrary dimensions. This web-method does not require any grid generation and can be implemented very efficiently. It yields smooth, high order accurate approximations with relatively low dimensional subspaces.

Bivariate box splines and smooth pp functions on a three direction mesh

Abstract Let S denote the space of bivariate piecewise polynomial functions of degree ⩽ k and smoothness ρ on the regular mesh generated by the three directions (1, 0), (0, 1), (1, 1). We construct a basis for S in terms of box splines and truncated powers. This allows us to determine the polynomials which are locally contained in S and to give upper and lower bounds for the degree of approximation. For ρ = ⌊ (2k − 2) 3 ⌋ , k ≢ 2 (3), the case of minimal degree k for given smoothness ρ, we identify the elements of minimal support in S and give a basis for S loc = {f ∈ S: supp f ⊆ Ω} , with Ω a convex subset of R 2 .

Geometric Hermite interpolation with maximal order and smoothness

Abstract We conjecture that, under suitable assumptions, splines of degree ⩽n can interpolate points on a smooth curve in R m with order of contact k − 1 = n − 1 + [ (n − 1) (m − 1) ] at every nth knot. Moreover, this Geometric Hermite Interpolant (GHI) has the optimal approximation order k + 1. We give a proof of this conjecture for planar quadratic spline curves and describe a simple construction of curvature continuous quadratic splines from control polygons.

Geometric Hermite interpolation

In this paper we present an overview over the more recent developments of Geometric Hermite Approximation Theory for planar curves. A general method to solve those problems is presented. Emphasis is put on the relations to differential geometry and to invariance against parameter transformations and the motion group of the underlying geometry. However, besides a few elementary cases, this leads to nonlinear systems of algebraic equations. Furthermore we give some geometric interpretations, a couple of examples and a detailed discussion of the case degree n=4 with one contact point.

Nonuniform web-splines

The construction of weighted extended B-splines (web-splines), as recently introduced by the authors and J. Wipper for uniform knot sequences, is generalized to the nonuniform case. We show that web-splines form a stable basis for splines on arbitrary domains in Rm which provides optimal approximation power. Moreover, homogeneous boundary conditions, as encountered frequently in finite element applications, can be satisfied exactly by using an appropriate weight function. To illustrate the performance of the method, it is applied to a scattered data fitting problem and a finite element approximation of an elliptic boundary value problem.

Introduction to the Web-method and its applications

Abstract The Web-method is a meshless finite element technique which uses weighted extended B-splines (Web-splines) on a tensor product grid as basis functions. It combines the computational advantages of B-splines and standard mesh-based elements. In particular, degree and smoothness can be chosen arbitrarily without substantially increasing the dimension. Hence, accurate approximations are obtained with relatively few parameters. Moreover, the regular grid is well suited for hierarchical refinement and multigrid techniques. This article should serve as an introduction to finite element approximation with B-splines. We first review the construction of Web-bases and discuss their basic properties. Then we illustrate the performance of Ritz–Galerkin schemes for a model problem and applications in linear elasticity. Finally, we discuss several implementation aspects.

Multigrid methods with web-splines

SummaryWe describe and analyze a multigrid algorithm for finite element approximations of second order elliptic boundary value problems with weightedextended b-splines (web-splines). This new technique provides high accuracy with relatively low-dimensional subspaces, does not require any grid generation, and is ideally suited for hierarchical solution techniques. In particular, we show that the standard W-cycle yields uniform convergence, i.e., the required number of iterations is bounded independent of the grid width.

A Diffusion Equation with a Nonmonotone Constitutive Function

We discuss the well-posedness of the model problem ut = o(ux)x on [0,1] x [0,T], T > 0, subject to given Neumann or Dirichlet boundary conditions at x = 0 and x = 1, and to the initial condition u(x,0) = f(x); the given functions f: [0,1] → R, o: R → R are assumed to be smooth, o(0) = 0, o satisfies the coercivity assumption Ęo(Ę) > cĘ2, for some constant c > 0 and for Ę Є R, and o is assumed to be decreasing on an interval (a,b) with a > 0. We present a recent nonuniqueness result in the special case when o is piecewise linear and study a related convexified problem.