Document Zbl 1211.65156

Fast simplicial finite element algorithms using Bernstein polynomials. (English) Zbl 1211.65156

Numer. Math. 117, No. 4, 631-652 (2011).

Summary: 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.

Cited in 1 Review

Cited in 26 Documents

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations
65Y20	Complexity and performance of numerical algorithms

Keywords:

numerical examples; algorithms; finite element; Bernstein basis; stiffness matrix; asymptotic complexity; mass matrices

Software:

DOLFIN

Cite Review PDF

Full Text: DOI

References:

[1]	Arnold D.N., Falk R.S., Winther R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006) · Zbl 1185.65204 · doi:10.1017/S0962492906210018
[2]	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) · Zbl 1227.65091 · doi:10.1016/j.cma.2008.12.017
[3]	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) · Zbl 0658.76001
[4]	Funaro, D.: Spectral elements for transport-dominated equations. In: Lecture Notes in Computational Science and Engineering, vol. 1. Springer-Verlag, Berlin (1997) · Zbl 0891.65118
[5]	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) · Zbl 1110.65093 · doi:10.1007/s10665-006-9085-7
[6]	Golub, G.H., Van Loan, C.F.: Matrix computations, 3rd edn. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1996) · Zbl 0865.65009
[7]	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) · Zbl 1014.78016 · doi:10.1006/jcph.2002.7118
[8]	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) · Zbl 1151.74419 · doi:10.1016/j.cma.2004.10.008
[9]	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) · Zbl 1116.76002
[10]	Lai, M.-J., Schumaker, L.L.: Spline functions on triangulations. In: Encyclopedia of Mathematics and its Applications, vol. 110. Cambridge University Press, Cambridge (2007) · Zbl 1185.41001
[11]	Logg A., Wells G.N.: DOLFIN: automated finite element computing. ACM Trans. Math. Softw. 37(20), 1–28 (2009) · Zbl 1364.65254 · doi:10.1145/1731022.1731030
[12]	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) · Zbl 1119.76043 · doi:10.1016/j.cma.2006.01.008
[13]	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) · Zbl 1140.76023 · doi:10.1002/fld.1711
[14]	Sevilla R., Fernández-Méndez S., Huerta A.: NURBS-enhanced finite element method (NEFEM)x. Int. J. Numer. Methods Eng. 76(1), 56–83 (2008) · Zbl 1162.65389 · doi:10.1002/nme.2311
[15]	Schumaker L.L.: Computing bivariate splines in scattered data fitting and the finite-element method. Numer. Algorithms 48(1–3), 237–260 (2008) · Zbl 1146.65019 · doi:10.1007/s11075-008-9175-x
[16]	Zienkiewicz, O.C., Taylor, R.L.: The finite element method, vol. 3, 5th edn. In: Fluid Dynamics. Butterworth-Heinemann, Oxford (2000) · Zbl 0991.74002

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.