Exponential Convergence of the hp Version of Isogeometric Analysis in 1D

Abstract

We establish exponential convergence of the hp-version of isogeometric analysis for second order elliptic problems in one spacial dimension. Specifically, we construct, for functions which are piecewise analytic with a finite number of algebraic singularities at a-priori known locations in the closure of the open domain Ω of interest, a sequence $(\varPi _{\sigma }^{\ell})_{\ell\geq 0}$ of interpolation operators which achieve exponential convergence. We focus on localized splines of reduced regularity so that the interpolation operators $(\varPi _{\sigma }^{\ell})_{\ell\geq 0}$ are Hermite type projectors onto spaces of piecewise polynomials of degree p ∼ ℓ whose differentiability increases linearly with p. As a consequence, the degree of conformity grows with N, so that asymptotically, the interpoland functions belong to C ^k(Ω) for any fixed, finite k. Extensions to two- and to three-dimensional problems by tensorization are possible.