Skip to main content
SpringerLink
Log in

Fast inversion of the simplicial Bernstein mass matrix

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

We consider the mass matrix arising from Bernstein polynomials as finite element shape functions. In particular, we give an explicit formula for its eigenvalues and exact characterization of the eigenspaces in terms of the Bernstein representation of orthogonal polynomials. We then derive a fast algorithm for solving linear systems involving the element mass matrix. After establishing these results, we describe the application of Bernstein techniques to the discontinuous Galerkin finite element method for hyperbolic conservation laws, obtaining optimal complexity algorithms. Finally, we present numerical results investigating the accuracy of the mass inversion algorithms and the scaling of total run-time for the function evaluation needed in DG time-stepping.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Ainsworth, M., Andriamaro, G., Davydov, O.: Bernstein-Bézier finite elements of arbitrary order and optimal assembly procedures. SIAM J. Sci. Comp. 33(6), 3087–3109 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ainsworth, M., Andriamaro, G., Davydov, O.: A Bernstein-Bézier basis for arbitrary order Raviart-Thomas finite elements. Construct. Approx. 41(1), 1–22 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, D.N., Falk, R.S., Winther, R.: Geometric decompositions and local bases for spaces of finite element differential forms. Comp. Methods Appl. Mechan. Eng. 198(21), 1660–1672 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bareiss, E.H.: Numerical solution of linear equations with Toeplitz and vector Toeplitz matrices. Numerische Mathematik 13(5), 404–424 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  5. Behnel, S., Bradshaw, R., Ewing, G.: Cython: C-extensions for Python (2008)

  6. Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, third edn. Springer, New York (2008)

  7. Cockburn, B., Shu, C.W.: The Runge-Kutta local projection \(P^1\)-discontinuous Galerkin finite element method for scalar conservation laws. RAIRO Modél. Math. Anal. Numér. 25(3), 337–361 (1991)

    MathSciNet  MATH  Google Scholar 

  8. Derriennic, M.M.: On multivariate approximation by Bernstein-type polynomials. J. Approx. Theory 45(2), 155–166 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dubiner, M.: Spectral methods on triangles and other domains. J. Sci. Comp. 6(4), 345–390 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Duffy, M.G.: Quadrature over a pyramid or cube of integrands with a singularity at a vertex. SIAM J. Num. Anal. 19(6), 1260–1262 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dumbser, M., Balsara, D.S., Toro, E.F., Munz, C.D.: A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227(18), 8209–8253 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Farouki, R.T., Goodman, T.N.T., Sauer, T.: Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains. Comp. Aided Geom. Design 20(4), 209–230 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43(1), 89–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hesthaven, J.S., Warburton, T.: Nodal high-order methods on unstructured grids: I. time-domain solution of Maxwell’s equations. J. Comput. Phys. 181(1), 186–221 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hesthaven, J.S., Warburton, T.: Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, vol. 54. Springer (2007)

  16. Hoteit, H., Ackerer, P., Mosé, R., Erhel, J., Philippe, B.: New two-dimensional slope limiters for discontinuous Galerkin methods on arbitrary meshes. Int. J. Num. Methods Eng. 61(14), 2566–2593 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jones, E., Oliphant, T., Peterson, P.: SciPy: Open source scientific tools for Python. http://www.scipy.org/ (2001)

  18. Karniadakis, G.E., Sherwin, S.J.: Spectral/\(hp\) element methods for computational fluid dynamics, 2nd edn. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2005)

  19. Kirby, R.C.: Fast simplicial finite element algorithms using Bernstein polynomials. Numerische Mathematik 117(4), 631–652 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kirby, R.C.: Low-complexity finite element algorithms for the de Rham complex on simplices. SIAM J. Sci. Comp. 36(2), A846–A868 (2014)

    Article  MathSciNet  Google Scholar 

  21. Kirby, R.C., Kieu, T.T.: Fast simplicial quadrature-based finite element operators using Bernstein polynomials. Numerische Mathematik 121(2), 261–279 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Klöckner, A., Warburton, T., Bridge, J., Hesthaven, J.S.: Nodal discontinuous Galerkin methods on graphics processors. J. Comput. Phys. 228(21), 7863–7882 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Koev, P.: Accurate computations with totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 29(3), 731–751 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lai, M.J., Schumaker, L.L.: Spline functions on triangulations, encyclopedia of mathematics and its applications, vol. 110. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  25. Levinson, N.: The Wiener RMS error criterion in filter design and prediction. J. Math. Phys. 25, 261–278 (1947)

    Article  MathSciNet  Google Scholar 

  26. Logg, A., Wells, G.N.: DOLFIN: automated finite element computing. ACM Trans. Math. Softw. 37(2): 20–28 (2010). doi:10.1145/1731022.1731030

  27. Shu, C.W.: Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comp. 9(6), 1073–1084 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  28. Strang, G.: Linear Algebra and Its Applications. Thomson, Brooks/Cole, Belmont (2006)

    MATH  Google Scholar 

  29. Stroud, A.H.: Approximate calculation of multiple integrals. Prentice-Hall Inc., Englewood Cliffs, N.J. Prentice-Hall Series in Automatic Computation (1971)

  30. Toro, E.F.: Riemann solvers and numerical methods for fluid dynamics, vol. 16. Springer (1999)

  31. Warburton, T.: private communication

  32. Warburton, T.: A low-storage curvilinear discontinuous Galerkin method for wave problems. SIAM J. Sci. Comp. 35(4), A1987–A2012 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhu, J., Qiu, J., Shu, C.W., Dumbser, M.: Runge-Kutta discontinuous Galerkin method using WENO limiters II: unstructured meshes. J. Comput. Phys. 227(9), 4330–4353 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhu, J., Zhong, X., Shu, C.W., Qiu, J.: Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes. J. Comput. Phys. 248, 200–220 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX, 76798-7328, USA

    Robert C. Kirby

Authors
  1. Robert C. Kirby
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert C. Kirby.

Additional information

The author acknowledges support from NSF grant CCF-1325480.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kirby, R.C. Fast inversion of the simplicial Bernstein mass matrix. Numer. Math. 135, 73–95 (2017). https://doi.org/10.1007/s00211-016-0795-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-016-0795-0

Keywords

Mathematics Subject Classification

Search

Navigation