<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml"><p>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 <italic>k</italic> 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 <italic>P5</italic> elements. We compare computational time and effort for several codes which are of high order in time and space and study their respective properties.</p></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 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.

Arbitrary High-Order Finite Element Schemes and High-Order Mass Lumping

Institut de Recherche Mathématique Avancée, UMR, 7501 de 1'ULP et du CNRS 7, rue René Descartes, 67084 Strasbourg Cedex, France

International Journal of Applied Mathematics and Computer Science

{"article-title":"Arbitrary High-Order Finite Element Schemes and High-Order Mass Lumping"}

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...

Arbitrary High-Order Finite Element Schemes and High-Order Mass Lumping

Publié en ligne: 11 oct. 2007

Pages: 375 - 393

DOI: https://doi.org/10.2478/v10006-007-0031-2

Mots clés
wave equation, finite element method, mass lumping, Cauchy-Kowalewski procedure

This content is open access.

Arbitrary High-Order Finite Element Schemes and High-Order Mass Lumping

Publié en ligne: 11 oct. 2007

Pages: 375 - 393

DOI: https://doi.org/10.2478/v10006-007-0031-2

Mots cléswave equation, finite element method, mass lumping, Cauchy-Kowalewski procedure

This content is open access.

Mots clés
wave equation, finite element method, mass lumping, Cauchy-Kowalewski procedure