Abstract
Fast algorithms for applying finite element mass and stiffness operators to the B-form of polynomials over d-dimensional simplices are derived. These rely on special properties of the Bernstein basis and lead to stiffness matrix algorithms with the same asymptotic complexity as tensor-product techniques in rectangular domains. First, special structure leading to fast application of mass matrices is developed. Then, by factoring stiffness matrices into products of sparse derivative matrices with mass matrices, fast algorithms are also obtained for stiffness matrices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold D.N., Falk R.S., Winther R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)
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)
Canuto, C., Yousuff Hussaini, M., Quarteroni, A., Zang, T.A.: Spectral methods in fluid dynamics. In: Springer Series in Computational Physics. Springer-Verlag, New York (1988)
Funaro, D.: Spectral elements for transport-dominated equations. In: Lecture Notes in Computational Science and Engineering, vol. 1. Springer-Verlag, Berlin (1997)
Giraldo F.X., Taylor M.A.: A diagonal-mass-matrix triangular-spectral-element method based on cubature points. J. Eng. Math. 56(3), 307–322 (2006)
Golub, G.H., Van Loan, C.F.: Matrix computations, 3rd edn. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1996)
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)
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)
Karniadakis, G.Em., Sherwin, S.J.: Spectral/hp element methods for computational fluid dynamics, 2nd edn. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2005)
Lai, M.-J., Schumaker, L.L.: Spline functions on triangulations. In: Encyclopedia of Mathematics and its Applications, vol. 110. Cambridge University Press, Cambridge (2007)
Logg A., Wells G.N.: DOLFIN: automated finite element computing. ACM Trans. Math. Softw. 37(20), 1–28 (2009)
Petersen S., Dreyer D., von Estorff O.: Assessment of finite and spectral element shape functions for efficient iterative simulations of interior acoustics. Comput. Methods Appl. Mech. Eng. 195, 6463–6478 (2006)
Sevilla R., Fernández-Méndez S., Huerta A.: NURBS-enhanced finite element method (NEFEM) for Euler equations. Int. J. Numer. Methods Fluids 57(9), 1051–1069 (2008)
Sevilla R., Fernández-Méndez S., Huerta A.: NURBS-enhanced finite element method (NEFEM)x. Int. J. Numer. Methods Eng. 76(1), 56–83 (2008)
Schumaker L.L.: Computing bivariate splines in scattered data fitting and the finite-element method. Numer. Algorithms 48(1–3), 237–260 (2008)
Zienkiewicz, O.C., Taylor, R.L.: The finite element method, vol. 3, 5th edn. In: Fluid Dynamics. Butterworth-Heinemann, Oxford (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work supported by the National Science Foundation under award number 0830655.
Rights and permissions
About this article
Cite this article
Kirby, R.C. Fast simplicial finite element algorithms using Bernstein polynomials. Numer. Math. 117, 631–652 (2011). https://doi.org/10.1007/s00211-010-0327-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-010-0327-2