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Exponential convergence of the \(hp\) version of isogeometric analysis in 1D. (English) Zbl 1282.65090

Azaïez, Mejdi (ed.) et al., Spectral and high order methods for partial differential equations – ICOSAHOM 2012. Selected papers from the ICOSAHOM conference, Gammarth, Tunisia, June 25–29, 2012. Cham: Springer (ISBN 978-3-319-01600-9/hbk; 978-3-319-01601-6/ebook). Lecture Notes in Computational Science and Engineering 95, 191-203 (2014).
Summary: 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 \(\varOmega\) 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 \(p\sim\ell\) whose differentiability increases linearly with \(p\). As a consequence, the degree of conformity grows with \(N\), so that asymptotically, the interpolant functions belong to \(C^k(\varOmega)\) for any fixed, finite \(k\). Extensions to two- and to three-dimensional problems by tensorization are possible.
For the entire collection see [Zbl 1279.65003].

MSC:

65L20 Stability and convergence of numerical methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
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