13 research outputs found
Ellipse-preserving Hermite interpolation and subdivision
We introduce a family of piecewise-exponential functions that have the
Hermite interpolation property. Our design is motivated by the search for an
effective scheme for the joint interpolation of points and associated tangents
on a curve with the ability to perfectly reproduce ellipses. We prove that the
proposed Hermite functions form a Riesz basis and that they reproduce
prescribed exponential polynomials. We present a method based on Green's
functions to unravel their multi-resolution and approximation-theoretic
properties. Finally, we derive the corresponding vector and scalar subdivision
schemes, which lend themselves to a fast implementation. The proposed vector
scheme is interpolatory and level-dependent, but its asymptotic behaviour is
the same as the classical cubic Hermite spline algorithm. The same convergence
properties---i.e., fourth order of approximation---are hence ensured
Isogeometric Analysis in advection-diffusion problems: tension splines approximation
We present a novel approach, within the new paradigm of isogeometric analysis
introduced by Hughes et al., to deal with advection dominated
advection-diffusion problems. The key ingredient is the use of Galerkin approximating
spaces of functions with high smoothness, as in IgA based on
classical B-splines, but particularly well suited to describe sharp layers involving
very strong gradients