An Introduction to the Numerical Analysis of Isogeometric Methods

Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 9)


This paper gives an introduction to isogeometric methods from a mathematical point of view, with special focus on some theoretical results that are part of the mathematical foundation of the method. The aim of this work is to serve as a complement to other existing references in the field, that are more engineering oriented, and to provide a reference that can be used for didactic purposes. We analyse variational techniques for the numerical resolutions of PDEs using isogeometric methods, that is, based on splines or NURBS, and we provide optimal approximation and error estimates for scalar elliptic problems. The theoretical results are demonstrated by some numerical examples. We also present the definition of structure-preserving discretizations with splines, a generalization of edge and face finite elements, also with approximation estimates and some numerical tests for time harmonic Maxwell equations in a cavity.


Isogeometric methods NURBS Finite elements De Rham complex 


  1. 1.
    Adams, R.A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65. Academic, New York/London (1975)Google Scholar
  2. 2.
    Apostolatos, A., Schmidt, R., Wüchner, R., Bletzinger, K.U.: A Nitsche-type formulation and comparison of the most common domain decomposition methods in isogeometric analysis. Int. J. Numer. Methods Eng. 97 (7), 473–504 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. (N.S.) 47 (2), 281–354 (2010)Google Scholar
  5. 5.
    Babuška, I., Strouboulis, T.: The Finite Element Method and Its Reliability. Numerical Mathematics and Scientific Computation. The Clarendon Press/Oxford University Press, New York (2001)zbMATHGoogle Scholar
  6. 6.
    Bazilevs, Y., Beirão da Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci. 16 (7), 1031–1090 (2006)Google Scholar
  7. 7.
    Beirão da Veiga, L., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for h-p-k-refinement in isogeometric analysis. Numer. Math. 118 (2), 271–305 (2011)Google Scholar
  8. 8.
    Beirão da Veiga, L., Buffa, A., Sangalli, G., Vázquez, R.: Mathematical analysis of variational isogeometric methods. Acta Numer. 23, 157–287 (2014)Google Scholar
  9. 9.
    Beirão da Veiga, L., Cho, D., Sangalli, G.: Anisotropic NURBS approximation in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 209–212, 1–11 (2012)Google Scholar
  10. 10.
    Boffi, D.: Finite element approximation of eigenvalue problems. Acta Numer. 19, 1–120 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Boffi, D., Fernandes, P., Gastaldi, L., Perugia, I.: Computational models of electromagnetic resonators: analysis of edge element approximation. SIAM J. Numer. Anal. 36 (4), 1264–1290 (electronic) (1999)Google Scholar
  12. 12.
    de Boor, C.: A Practical Guide to Splines. Applied Mathematical Sciences, vol. 27, revised edn. Springer, New York (2001)Google Scholar
  13. 13.
    Borden, M.J., Scott, M.A., Evans, J.A., Hughes, T.J.R.: Isogeometric finite element data structures based on Bézier extraction of NURBS. Int. J. Numer. Methods Eng. 87 (1–5), 15–47 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Buffa, A., Cho, D., Kumar, M.: Characterization of T-splines with reduced continuity order on T-meshes. Comput. Methods Appl. Mech. Eng. 201–204, 112–126 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Buffa, A., Rivas, J., Sangalli, G., Vázquez, R.: Isogeometric discrete differential forms in three dimensions. SIAM J. Numer. Anal. 49 (2), 818–844 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Buffa, A., Sangalli, G., Vázquez, R.: Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations. J. Comput. Phys. 257, Part B, 1291–1320 (2014)Google Scholar
  17. 17.
    Buffa, A., Vázquez, R., Sangalli, G., Beirão da Veiga, L.: Approximation estimates for isogeometric spaces in multipatch geometries. Numer. Methods Partial Differ. Equ. 31 (2), 422–438 (2015)Google Scholar
  18. 18.
    Cohen, E., Riesenfeld, R., Elber, G.: Geometric Modeling with Splines: An Introduction. AK Peters, Wellesley (2001)zbMATHGoogle Scholar
  19. 19.
    Collier, N., Dalcin, L., Pardo, D., Calo, V.M.: The cost of continuity: performance of iterative solvers on isogeometric finite elements. SIAM J. Sci. Comput. 35 (2), A767–A784 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cottrell, J.A., Hughes, T., Reali, A.: Studies of refinement and continuity in isogeometric structural analysis. Comput. Methods Appl. Mech. Eng. 196, 4160–4183 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester/Hoboken (2009)CrossRefGoogle Scholar
  22. 22.
    Dalcin, L., Collier, N., Vignal, P., Cortes, A., Calo, V.: PetIGA: a framework for high-performance isogeometric analysis (2015). arXiv:1305.4452v3 [cs.MS]Google Scholar
  23. 23.
    Dauge, M.: Benchmark computations for Maxwell equations for the approximation of highly singular solutions. (2014)
  24. 24.
    Dörfel, M., Jüttler, B., Simeon, B.: Adaptive isogeometric analysis by local h-refinement with T-splines. Comput. Methods Appl. Mech. Eng. 199 (5–8), 264–275 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    de Falco, C., Reali, A., Vázquez, R.: GeoPDEs: a research tool for isogeometric analysis of PDEs. Adv. Eng. Softw. 42 (12), 1020–1034 (2011)CrossRefzbMATHGoogle Scholar
  26. 26.
    Govindjee, S., Strain, J., Mitchell, T.J., Taylor, R.L.: Convergence of an efficient local least-squares fitting method for bases with compact support. Comput. Methods Appl. Mech. Eng. 213–216, 84–92 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194 (39–41), 4135–4195 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hughes, T.J.R., Reali, A., Sangalli, G.: Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of p-method finite elements with k-method NURBS. Comput. Methods Appl. Mech. Eng. 197 (49–50), 4104–4124 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kellogg, R.B.: On the Poisson equation with intersecting interfaces. Appl. Anal. 4 (2), 101–129 (1974/75)Google Scholar
  31. 31.
    Kiendl, J., Bazilevs, Y., Hsu, M.C., Wüchner, R., Bletzinger, K.U.: The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches. Comput. Methods Appl. Mech. Eng. 199 (37–40), 2403–2416 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kleiss, S.K., Pechstein, C., Jüttler, B., Tomar, S.: IETI-isogeometric tearing and interconnecting. Comput. Methods Appl. Mech. Eng. 247–248, 201–215 (2012)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Lee, B.G., Lyche, T., Mørken, K.: Some examples of quasi-interpolants constructed from local spline projectors. In: Mathematical Methods for Curves and Surfaces, Oslo, 2000. Innovations in Applied Mathematics, pp. 243–252. Vanderbilt University Press, Nashville (2001)Google Scholar
  34. 34.
    Monk, P.: Finite Element Methods for Maxwell’s Equations. Oxford University Press, Oxford (2003)CrossRefzbMATHGoogle Scholar
  35. 35.
    Morin, P., Nochetto, R.H., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44 (4), 631–658 (2002/2003)Google Scholar
  36. 36.
    Nédélec, J.C.: Mixed finite elements in \(\mathbb{R}^{3}\). Numer. Math. 35, 315–341 (1980)Google Scholar
  37. 37.
    Nguyen, V.P., Bordas, S.P.A., Rabczuk, T.: Isogeometric analysis: an overview and computer implementation aspects (2013). arXiv:1205.2129v2 [cs.NA]Google Scholar
  38. 38.
    Nguyen, V.P., Kerfriden, P., Brino, M., Bordas, S.P.A., Bonisoli, E.: Nitsche’s method for two and three dimensional NURBS patch coupling. Comput. Mech. 53 (6), 1163–1182 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Pauletti, M.S., Martinelli, M., Cavallini, N., Antolín, P.: Igatools: an isogeometric analysis library. SIAM J. Sci. Comput. 37 (4), C465–C496 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Petzoldt, M.: Regularity and error estimators for elliptic problems with discontinuous coefficients. Ph.D. thesis, Freie Universität Berlin (2001)Google Scholar
  41. 41.
    Piegl, L., Tiller, W.: The Nurbs Book. Springer, New York (1997)CrossRefzbMATHGoogle Scholar
  42. 42.
    Ratnani, A., Sonnendrücker, E.: An arbitrary high-order spline finite element solver for the time domain Maxwell equations. J. Sci. Comput. 51, 87–106 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Rogers, D.F.: An Introduction to NURBS: With Historical Perspective. Morgan Kaufmann Publishers Inc., San Francisco (2001)Google Scholar
  44. 44.
    Ruess, M., Schillinger, D., Özcan, A.I., Rank, E.: Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries. Comput. Methods Appl. Mech. Eng. 269, 46–71 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Sabin, M.A.: Spline finite elements. Ph.D. thesis, Cambridge University (1997)Google Scholar
  46. 46.
    Schumaker, L.L.: Spline Functions: Basic Theory. Cambridge Mathematical Library, 3rd edn. Cambridge University Press, Cambridge (2007)Google Scholar
  47. 47.
    Scott, M.: T-splines as a design-through-analysis technology. Ph.D. thesis, The University of Texas at Austin (2011)Google Scholar
  48. 48.
    Speleers, H., Manni, C., Pelosi, F., Sampoli, M.L.: Isogeometric analysis with Powell-Sabin splines for advection-diffusion-reaction problems. Comput. Methods Appl. Mech. Eng. 221–222, 132–148 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Takacs, T., Jüttler, B.: Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis. Comput. Methods Appl. Mech. Eng. 200 (49–52), 3568–3582 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Takacs, T., Jüttler, B.: Regularity properties of singular parameterizations in isogeometric analysis. Graph. Models 74 (6), 361–372 (2012)CrossRefGoogle Scholar
  51. 51.
    Thomas, D., Scott, M., Evans, J., Tew, K., Evans, E.: Bézier projection: a unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis. Comput. Methods Appl. Mech. Eng. 284, 55–105 (2015)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Dipartimento di Matematica e ApplicazioniUniversità degli Studi di Milano-BicoccaMilanoItaly
  2. 2.Istituto di Matematica Applicata e Tecnologie Informatiche ‘E. Magenes’ del CNRPaviaItaly
  3. 3.Dipartimento di MatematicaUniversità di PaviaPaviaItaly

Personalised recommendations