Arbitrary high-order finite element schemes and high-order mass lumping
Sébastien Jund; Stéphanie Salmon
International Journal of Applied Mathematics and Computer Science (2007)
- Volume: 17, Issue: 3, page 375-393
- ISSN: 1641-876X
Access Full Article
top
Abstract
topHow to cite
topJund, Sébastien, and Salmon, Stéphanie. "Arbitrary high-order finite element schemes and high-order mass lumping." International Journal of Applied Mathematics and Computer Science 17.3 (2007): 375-393. <http://eudml.org/doc/207844>.
@article{Jund2007,
abstract = {Computers are becoming sufficiently powerful to permit to numerically solve problems such as the wave equation with high-order methods. In this article we will consider Lagrange finite elementsof order k and show how it is possible to automatically generate the mass and stiffness matrices of any order with the help of symbolic computation software. We compare two high-order time discretizations: an explicit one using a Taylor expansion in time (a Cauchy-Kowalewski procedure) and an implicit Runge-Kutta scheme. We also construct in a systematic way a high-order quadrature which is optimal in terms of the number of points, which enables the use of mass lumping, up to P5 elements. We compare computational time and effort for several codes which are of high order in time and space and study their respective properties.},
author = {Jund, Sébastien, Salmon, Stéphanie},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {wave equation; finite element method; Cauchy-Kowalewski procedure; mass lumping; symbolic computation; higher order approximation},
language = {eng},
number = {3},
pages = {375-393},
title = {Arbitrary high-order finite element schemes and high-order mass lumping},
url = {http://eudml.org/doc/207844},
volume = {17},
year = {2007},
}
TY - JOUR
AU - Jund, Sébastien
AU - Salmon, Stéphanie
TI - Arbitrary high-order finite element schemes and high-order mass lumping
JO - International Journal of Applied Mathematics and Computer Science
PY - 2007
VL - 17
IS - 3
SP - 375
EP - 393
AB - Computers are becoming sufficiently powerful to permit to numerically solve problems such as the wave equation with high-order methods. In this article we will consider Lagrange finite elementsof order k and show how it is possible to automatically generate the mass and stiffness matrices of any order with the help of symbolic computation software. We compare two high-order time discretizations: an explicit one using a Taylor expansion in time (a Cauchy-Kowalewski procedure) and an implicit Runge-Kutta scheme. We also construct in a systematic way a high-order quadrature which is optimal in terms of the number of points, which enables the use of mass lumping, up to P5 elements. We compare computational time and effort for several codes which are of high order in time and space and study their respective properties.
LA - eng
KW - wave equation; finite element method; Cauchy-Kowalewski procedure; mass lumping; symbolic computation; higher order approximation
UR - http://eudml.org/doc/207844
ER -
References
top- Butcher J.C. (2003): Numerical Methods for Ordinary Differential Equations. Chichester: John Wiley and Sons. Zbl1040.65057
- Chin-Joe-Kong M.J.S., Mulder W.A. and Van Veeldhuizen M. (1999): Higher-order triangular and tetrahedral finite elements with mass lumping for solving the wave equation. Journal of Engineering Mathematics Vol.35, No.4, pp.405-426. Zbl0948.74057
- Ciarlet P.G. (1978): The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam. Zbl0383.65058
- Cohen G. (2002): Higher Order Numerical Method for Transient Wave Equations. Berlin: Springer-Verlag. Zbl0985.65096
- Cohen G., Joly P., Roberts J.E. and Tordjman N. (2001): Higher order triangular finite element with mass lumping forthe wave equation. SIAM Journal on Numerical Analysis, Vol.38, No.6, pp.2047-2078. Zbl1019.65077
- Cohen G., Joly P. and Tordjman N.(1994): Higher-order finite elements with mass lumping for the 1D wave equation. Finite Elements in Analysis and Design, Vol.16 , No.3-4, pp.329-336. Zbl0865.65072
- Cohen G., Joly P. and Tordjman N. (1993): Construction and analysis of higher order finite elements with mass lumping for the wave equation, In: Proceedings of the Second International Conference on Mathematical and Numerical Aspects of Wave Propagation Phenomena, SIAM, Philadelphia, pp.152-160. Zbl0814.65096
- Dumbser M. (2005): Arbitrary High Order Schemes for the Solution of Hyperbolic Conservation Laws in Complex Domains. Ph.D. thesis, Stuttgart University.
- Fix G.J. (1972): Effect of quadrature errors in the finite element approximation of steady state, eigenvalue and parabolic problems, In: The Mathematical Foundations of the Finite Element Method with Applications to the Partial Differential Equations, (A.K. Aziz, Ed.), New York: Academic Press, pp.525-556.
- Lax P.D. and Wendroff B. (1960): Systems of conservation laws. Communications on Pure Applied Mathematics, Vol.13, pp.217-237. Zbl0152.44802
- Mulder W.A. (1996): A comparison between higher-order finite elements and finite differences for solving the wave equation, In: Proceedings of the Second ECCOMAS Conference Numerical Methods in Engineering, (J.-A. Désidéri, P. Le Tallec, E. Onate, J. Périaux and E. Stein, Eds.), Chichester: John Wiley and Sons, pp.344-350.
- Tordjman N. (1995): Eléments finis d'ordre élevés avec condensation de masse pour l'équation des onde. Ph.D. Thesis, Université Paris IX Dauphine, Paris.
- Tarev V.A. and Toro E.F. (2002): ADER: Arbitrary high order Godunov approach. Journal of Scientific Computing, Vol.17, No.1-4, pp.609-618 Zbl1024.76028
NotesEmbed ?
topYou must be logged in to post comments.
To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.