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Mathematical analysis of variational isogeometric methods*

Published online by Cambridge University Press:  12 May 2014

L. Beirão da Veiga,
A. Buffa,
G. Sangalli and
R. Vázquez
L. Beirão da Veiga
Affiliation:
Dipartimento di Matematica, Università di Milano, via Saldini 50, 20133, Milano, Italy, E-mail: lourenco.beirao@unimi.itIstituto di Matematica Applicata e Tecnologie Informatiche, ‘E. Magenes’ del CNR, via Ferrata 1, 27100, Pavia, Italy, E-mail: {annalisa.buffa, vazquez}@imati.cnr.it
A. Buffa
Affiliation:
Istituto di Matematica Applicata e Tecnologie Informatiche, ‘E. Magenes’ del CNR, via Ferrata 1, 27100, Pavia, Italy, E-mail: {annalisa.buffa, vazquez}@imati.cnr.it
G. Sangalli
Affiliation:
Istituto di Matematica Applicata e Tecnologie Informatiche, ‘E. Magenes’ del CNR, via Ferrata 1, 27100, Pavia, Italy, E-mail: {annalisa.buffa, vazquez}@imati.cnr.itDipartimento di Matematica, Università di Pavia, via Ferrata 1, 27100, Pavia, Italy, E-mail: giancarlo.sangalli@unipv.it
R. Vázquez
Affiliation:
Istituto di Matematica Applicata e Tecnologie Informatiche, ‘E. Magenes’ del CNR, via Ferrata 1, 27100, Pavia, Italy, E-mail: {annalisa.buffa, vazquez}@imati.cnr.it
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This review paper collects several results that form part of the theoretical foundation of isogeometric methods. We analyse variational techniques for the numerical resolution of PDEs based on splines or NURBS and we provide optimal approximation and error estimates in several cases of interest. The theory presented also includes estimates for T-splines, which are an extension of splines allowing for local refinement. In particular, we focus our attention on elliptic and saddle point problems, and we define spline edge and face elements. Our theoretical results are demonstrated by a rich set of numerical examples. Finally, we discuss implementation and efficiency together with preconditioning issues for the final linear system.

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Type
Research Article
Information
Acta Numerica , Volume 23 , May 2014, pp. 157 - 287
Copyright
Copyright © Cambridge University Press 2014 

Footnotes

*

Colour online for monochrome figures available at journals.cambridge.org/anu.

References

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