Geodesic
Arbitrary high-order finite element schemes and high-order mass lumping
International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) no. 3, pp. 375-393.

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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 elements of 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.
Keywords: wave equation, finite element method, mass lumping, Cauchy-Kowalewski procedure
Mots-clés : równanie falowe, metoda elementów skończonych, twierdzenie Cauchy'ego-Kowalevskiego 
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Jund, S.; Salmon, S. Arbitrary high-order finite element schemes and high-order mass lumping. International Journal of Applied Mathematics and Computer Science, Tome 17 (2007) no. 3, pp. 375-393. http://geodesic.mathdoc.fr/item/IJAMCS_2007_17_3_a7/

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