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Bernstein-Bézier techniques for divergence of polynomial spline vector fields in ℝn

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Abstract

Bernstein-Bézier techniques for analyzing polynomial spline fields in n variables and their divergence are developed. Dimension and a minimal determining set for continuous piecewise divergence-free spline fields on the Alfeld split of a simplex in ℝn are obtained using the new techniques, as well as the dimension formula for continuous piecewise divergence-free splines on the Alfeld refinement of an arbitrary simplicial partition in ℝn.

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References

  1. Ainsworth, M., Andriamaro, G., Davydov, O.: A Bernstein-Bézier basis for arbitrary order Raviart-Thomas finite elements. Constr. Approx. 41, 1–22 (2014)

    Article  MATH  Google Scholar 

  2. Boffi, D., Brezzi, F., Demkowicz, L.F., Durn, R.G., Falk, R.S., Fortin, M.: Mixed Finite Elements, Compatibility Conditions, and Applications, Springer (2008)

  3. Alfeld, P.: MDS software, http://www.math.utah.edu/~pa

  4. Alfeld, P., Sorokina, T.: Linear differential operators on bivariate spline spaces and spline vector fields. BIT Numer. Math. 56(1), 15–32 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Billera, L.J., Haas, R.: The dimension and bases of divergence-free splines; a homological approach. Approx. Theory Appl. 7, 91–99 (1991)

    MathSciNet  MATH  Google Scholar 

  6. De Boor, C.: B-form basics, Geometric modeling: Algorithms and new trends. In: Farin, G.E. (ed.) SIAM Publications, Philadelphia (1987)

  7. Chui, C., Lai, M.-J.: Multivariate vertex splines and finite elements. J. Approx. Theory 60, 245–343 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lai, M.-J., Schumaker, L. L.: Spline functions on triangulations. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  9. Stenberg, R.: Analysis of mixed finite elements methods for the Stokes problem: a unified approach. Math. Comp. 42(165), 9–23 (1984)

    MathSciNet  MATH  Google Scholar 

  10. Zhang, S.: Quadratic divergence-free finite elements on PowellSabin tetrahedral grids. Calcolo 48(3), 211–244 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We would like to thank the referees for their suggestions, and Peter Alfeld for useful discussions, and for adopting his spline software [3] to spline vector fields.

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Authors and Affiliations

  1. Towson University, 7800 York Road, Towson, MD, 21252, USA

    Tatyana Sorokina

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  1. Tatyana Sorokina
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Correspondence to Tatyana Sorokina.

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Communicated by: Alexander Barnett

This work was partially supported by agrant from the Simons Foundation (#235411 to Tatyana Sorokina)

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Sorokina, T. Bernstein-Bézier techniques for divergence of polynomial spline vector fields in ℝn . Adv Comput Math 44, 227–244 (2018). https://doi.org/10.1007/s10444-017-9541-0

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  • DOI: https://doi.org/10.1007/s10444-017-9541-0

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