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Fast simplicial quadrature-based finite element operators using Bernstein polynomials

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Abstract

We derive low-complexity matrix-free finite element algorithms for simplicial Bernstein polynomials on simplices. Our techniques, based on a sparse representation of differentiation and special block structure in the matrices evaluating B-form polynomials at warped Gauss points, apply to variable coefficient problems as well as constant coefficient ones, thus extending our results in Kirby (Numer Math, 2011, in press).

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References

  1. Ainsworth, M., Andriamaro, G., Davydov, O.: Bernstein–Bézier finite elements of arbitrary order and optimal assembly procedures. SIAM J. Scientific Comput. (2011, in press)

  2. Martinez, J.-J., Marco, A.: Accurate numerical linear algebra with Bernstein–Vandermonde matrices (2008). arXiv:0812.3115v1

  3. Arnold D.N., Falk R.S., Winther R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold D.N., Falk R.S., Winther R.: Geometric decompositions and local bases for spaces of finite element differential forms. Comput. Methods Appl. Mech. Eng. 198, 1660–1672 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Awanou G., Lai M.J.: Trivariate spline approximations of 3d navier–stokes equations. Math. Comput. 74, 585–602 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Awanou G., Lai M.J., Wenston P.: The multivariate spline method for numerical solution of partial differential equations and scattered data interpolation. In: Chen, G., Lai, M.J. (eds) Wavelets and Splines: Athens 2005, pp. 24–74. Nashboro Press, Brentwood (2005)

    Google Scholar 

  7. Bochev, P., Ridzal, D., Peterson, K.: Intrepid: interoperable tools for compatible discretizations. http://trilinos.sandia.gov/packages/intrepid

  8. Brenner S.C., Ridgway Scott L.: The Mathematical Theory of Finite Element Methods, vol. 15. Texts in Applied Mathematics, 3rd edn. Springer, New York (2008)

    Book  Google Scholar 

  9. Canuto C., Yousuff Hussaini M., Quarteroni A., Zang T.A.: Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics. Springer, New York (1988)

    Google Scholar 

  10. Funaro D.: Spectral Elements for Transport-Dominated Equations, vol. 1. Lecture Notes in Computational Science and Engineering. Springer, Berlin (1997)

    Book  Google Scholar 

  11. Heroux M.A., Bartlett R.A., Howle V.E., Hoekstra R.J., Hu J.J., Kolda T.G., Lehoucq R.B., Long K.R., Pawlowski R.P., Phipps E.T., Salinger A.G., Thornquist H.K., Tuminaro R.S., Willenbring J.M., Williams A., Stanley K.S.: An overview of the trilinos project. ACM Trans. Math. Softw. 31, 397–423 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hu X.L., Han D.F., Lai M.J.: Bivariate splines of various degrees for numerical solution of partial differential equations. SIAM J. Scientific Comput. 29, 1338 (2008)

    Article  MathSciNet  Google Scholar 

  13. Karniadakis G.E., Sherwin S.J.: Spectral/hp Element Methods for Computational Fluid Dynamics Numerical Mathematics and Scientific Computation, 2nd edn. Oxford University Press, New York (2005)

    Book  Google Scholar 

  14. Kirby, R.C., Long, K., van Bloemen Waanders, B.: Unified embedded parallel finite element computations via software-based Fréchet differentiation. SIAM J. Scientific Comput. (2011, in press)

  15. Kirby, R.C.: Fast application of some finite element bilinear forms using bernstein polynomials. Numer. Math. (2011, in press)

  16. Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29, 731–751 (2007, electronic)

    Google Scholar 

  17. Lai M.-J., Schumaker L.L.: Spline Functions on Triangulations, vol. 110. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  18. Long K.: Chapter contribution. In: Biegler, L., Ghattas, O., Heinkenschloss, M., Bloemen Waanders, B. (eds) Large-Scale PDE-Constrained Optimization, vol. 30. Lecture Notes in Computational Science and Engineering, Springer, Berlin (2003)

    Google Scholar 

  19. 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 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Texas Tech University, PO Box 1042, Lubbock, TX, 79409-1042, USA

    Robert C. Kirby & Kieu Tri Thinh

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  1. Robert C. Kirby
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  2. Kieu Tri Thinh
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Correspondence to Robert C. Kirby.

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Work supported by the National Science Foundation under award number 0830655.

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Kirby, R.C., Thinh, K.T. Fast simplicial quadrature-based finite element operators using Bernstein polynomials. Numer. Math. 121, 261–279 (2012). https://doi.org/10.1007/s00211-011-0431-y

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  • DOI: https://doi.org/10.1007/s00211-011-0431-y

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