Abstract
We develop a Hermite interpolation scheme and prove error bounds for \(C^1\) bivariate piecewise polynomial spaces of Argyris type vanishing on the boundary of curved domains enclosed by piecewise conics.
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J. Bloomenthal et al., Introduction to Implicit Surfaces (Morgan-Kaufmann Publishers Inc., San Francisco, 1997)
K. Böhmer, On finite element methods for fully nonlinear elliptic equations of second order. SIAM J. Numer. Anal. 46, 1212–1249 (2008)
K. Böhmer, Numerical Methods for Nonlinear Elliptic Differential Equations: A Synopsis (Oxford University Press, Oxford, 2010)
S.C. Brenner, L.R. Scott, The Mathematical Theory of Finite Element Methods (Springer, New York, 1994)
P.G. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978)
O. Davydov, Smooth finite elements and stable splitting, Berichte “Reihe Mathematik” der Philipps-Universität Marburg, 2007-4 (2007). An adapted version has appeared as [3, Sect. 4.2.6]
O. Davydov, A. Saeed, Stable splitting of bivariate spline spaces by Bernstein-Bézier methods, in Curves and Surfaces - 7th International Conference, Avignon, France, June 24–30, 2010, eds. by J.-D. Boissonnat et al. LNCS, vol. 6920 (Springer, Berlin, 2012), pp. 220–235
O. Davydov, A. Saeed, Numerical solution of fully nonlinear elliptic equations by Böhmer’s method. J. Comput. Appl. Math. 254, 43–54 (2013)
O. Davydov, W.P. Yeo, Macro-element hierarchical Riesz bases, in Mathematical Methods for Curves and Surfaces: 8th International Conference, Oslo, 2012, eds. by M. Floater et al. LNCS, vol. 8177 (Springer, Berlin, 2014), pp. 112–134
O. Davydov, A. Saeed, \(C^1\) quintic splines on domains enclosed by piecewise conics and numerical solution of fully nonlinear elliptic equations. Appl. Numer. Math. 116, 172–183 (2017)
O. Davydov, G. Nürnberger, F. Zeilfelder, Bivariate spline interpolation with optimal approximation order. Constr. Approx. 17, 181–208 (2001)
O. Davydov, G. Kostin, A. Saeed, Polynomial finite element method for domains enclosed by piecewise conics. CAGD 45, 48–72 (2016)
K. Höllig, U. Reif, J. Wipper, Weighted extended B-spline approximation of Dirichlet problem. SIAM J. Numer. Anal. 39(2), 442–462 (2001)
T.J.R. Hughes, J.A. Cottrel, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)
M.J. Lai, L.L. Schumaker, Spline Functions on Triangulations (Cambridge University Press, Cambridge, 2007)
L.L. Schumaker, On super splines and finite elements. SIAM J. Numer. Anal. 26(4), 997–1005 (1989)
Acknowledgements
This research has been supported in part by the grant UBD/PNC2/2/RG/1(301) from Universiti Brunei Darussalam.
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Davydov, O., Yeo, W.P. (2017). Approximation by \(C^1\) Splines on Piecewise Conic Domains. In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XV: San Antonio 2016. AT 2016. Springer Proceedings in Mathematics & Statistics, vol 201. Springer, Cham. https://doi.org/10.1007/978-3-319-59912-0_2
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DOI: https://doi.org/10.1007/978-3-319-59912-0_2
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