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.
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Acknowledgements
A. Buffa and G. Sangalli were partially supported by the European Research Council through the FP7 Ideas Starting Grant n. 205004, by the Italian MIUR through the FIRB Grant RBFR08CZ0S and by the European Commission through the FP7 Factories of the Future project TERRIFIC. C. Schwab was supported by the European Research Council through the FP7 Ideas Advanced Grant n. 247277. This support is gratefully acknowledged.
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Buffa, A., Sangalli, G., Schwab, C. (2014). Exponential Convergence of the hp Version of Isogeometric Analysis in 1D. In: Azaïez, M., El Fekih, H., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012. Lecture Notes in Computational Science and Engineering, vol 95. Springer, Cham. https://doi.org/10.1007/978-3-319-01601-6_15
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DOI: https://doi.org/10.1007/978-3-319-01601-6_15
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