Skip to main content
Log in

Spline spaces on TR-meshes with hanging vertices

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

Polynomial spline spaces defined on mixed meshes consisting of triangles and rectangles are studied for the C 0 case. These include triangulations with hanging vertices as well as T-meshes. In addition to dimension formulae, explicit basis functions are constructed, and their supports and stability are discussed. The approximation power of the spaces is also treated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ainsworth M., Oden J.T.: A posteriori error estimation in finite element analysis. Wiley-Interscience, New York (2000)

    MATH  Google Scholar 

  2. Brix K., Pinto M.C., Dahmen W.: A multilevel preconditioner for the interior penalty discontinuous Galerkin method. SIAM J. Numer. Anal. 46, 2742–2768 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carstensen C., Hu J.: Hanging nodes in the unifying theory of a posteriori finite element error control. J. Comp. Math. 27, 215–236 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Deng J.-S., Chen F-L., Feng Y.-Y.: Dimensions of spline spaces over T-meshes. J. Comput. Appl. Math. 194, 267–283 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Deng J.-S., Chen F-L., Li X., Hu C., Tong W., Yang Z., Feng Y.: Polynomial splines over heirachical T-meshes. Graphical Mod. 74, 76–86 (2008)

    Article  Google Scholar 

  6. Dörfel M. R., Jüttler B., Simeon B.: Adaptive isogeometric analysis by local h-refinements with T-splines. Comp. Meth. Appl. Mech. Eng. 199, 264–275 (2010)

    Article  Google Scholar 

  7. Forsey D., Bartels R.: Hierarchical B-spline refinement. Comput. Graph. 22(4), 205–212 (1988)

    Article  Google Scholar 

  8. Goël J.J.: Construction of basic functions for numerical utilization of Ritz’s method. Numer. Math. 12, 435–447 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ibrahim A.Kh.: Dimension of superspline spaces defined over rectilinear partitions. In: Chui, C., Schumaker, L., Ward, J. (eds) Approximation Theory VI, pp. 337–340. Academic Press, New York (1989)

    Google Scholar 

  10. Lai M.J., Schumaker L.L.: Spline Functions on Triangulations. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  11. Li C.J., Wang R.H., Zhang F.: Improvement on the dimensions of splines on T-meshes. J. Inf. Comput. Sci. 3, 235–244 (2006)

    MATH  Google Scholar 

  12. Prautzsch H., Boehm W., Paluszny M.: Bézier and B-spline Techniques. Springer, Berlin (2002)

    MATH  Google Scholar 

  13. Schumaker L.L.: Computing bivariate splines in scattered data fitting and the finite element method. Numer. Algorithms 48, 237–260 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Šolın P., Červený J., Doležel I.: Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM. Math. Comput. Simul. 77, 117–132 (2008)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Larry L. Schumaker.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schumaker, L.L., Wang, L. Spline spaces on TR-meshes with hanging vertices. Numer. Math. 118, 531–548 (2011). https://doi.org/10.1007/s00211-010-0353-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-010-0353-0

Mathematics Subject Classification (2000)

Navigation