Summary
This paper considers the h p-finite element discretization of an elliptic boundary value problem using tetrahedral elements. The discretization uses a polynomial basis in which the number of nonzero entries per row is bounded independently of the polynomial degree. The authors present an algorithm which computes the nonzero entries of the stiffness matrix in optimal complexity. The algorithm is based on sum factorization and makes use of the nonzero pattern of the stiffness matrix.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
Milton Abramowitz and Irene A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964.
I. Babuska, M. Griebel, and J. Pitkäranta. The problem of selecting the shape functions for a p-type finite element. Internat. J. Numer. Methods Engrg., 28(8):1891–1908, 1989.
S. Beuchler and V. Pillwein. Sparse shape functions for tetrahedral p-FEM using integrated Jacobi polynomials. Computing, 80(4):345–375, 2007.
S. Beuchler, V. Pillwein, and S. Zaglmayr. Sparsity optimized high order finite element functions for h(d i v) on simplices. Technical Report 2010-07, RICAM, 2010. submitted.
S. Beuchler and J. Schöberl. New shape functions for triangular p-FEM using integrated Jacobi polynomials. Numer. Math., 103(3):339–366, 2006.
Leszek Demkowicz, Jason Kurtz, David Pardo, Maciej Paszyński, Waldemar Rachowicz, and Adam Zdunek. Computing with hp-adaptive finite elements. Vol. 2. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, FL, 2008. Frontiers: three dimensional elliptic and Maxwell problems with applications.
George Em Karniadakis and Spencer J. Sherwin. Spectral/hp element methods for computational fluid dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, second edition, 2005.
J.M. Melenk, K. Gerdes, and C. Schwab. Fully discrete h p-finite elements: Fast quadrature. Comp. Meth. Appl. Mech. Eng., 190:4339–4364, 1999.
Ch. Schwab. p- and hp-finite element methods. Numerical Mathematics and Scientific Computation. The Clarendon Press Oxford University Press, New York, 1998. Theory and applications in solid and fluid mechanics.
Acknowledgements
The work has been supported by the FWF projects P20121, P20162, and P23484.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Beuchler, S., Pillwein, V., Zaglmayr, S. (2013). Fast Summation Techniques for Sparse Shape Functions in Tetrahedral hp-FEM. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds) Domain Decomposition Methods in Science and Engineering XX. Lecture Notes in Computational Science and Engineering, vol 91. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35275-1_60
Download citation
DOI: https://doi.org/10.1007/978-3-642-35275-1_60
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-35274-4
Online ISBN: 978-3-642-35275-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)