# Higher Order Triangular Finite Elements with Mass Lumping for the Wave Equation

@article{Cohen2001HigherOT,
title={Higher Order Triangular Finite Elements with Mass Lumping for the Wave Equation},
author={Gary Cohen and Patrick Joly and Jean E. Roberts and Nathalie Tordjman},
journal={SIAM J. Numer. Anal.},
year={2001},
volume={38},
pages={2047-2078}
}
• G. Cohen, +1 author N. Tordjman
• Published 1 November 2000
• Computer Science, Mathematics
• SIAM J. Numer. Anal.
In this article, we construct new higher order finite element spaces for the approximation of the two-dimensional (2D) wave equation. These elements lead to explicit methods after time discretization through the use of appropriate quadrature formulas which permit mass lumping. These formulas are constructed explicitly. Error estimates are provided for the corresponding semidiscrete problem. Finally, higher order finite difference time discretizations are proposed and various numerical results… Expand
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#### References

SHOWING 1-10 OF 45 REFERENCES
Higher-order finite elements with mass-lumping for the 1D wave equation
• Mathematics
• 1994
Abstract This paper is devoted to the construction and analysis of a method, higher order in space and time, for solving the one-dimensional wave equation. This method is based on P3 Lagrange finiteExpand
A New Family of Mixed Finite Elements for the Linear Elastodynamic Problem
• Computer Science, Mathematics
• SIAM J. Numer. Anal.
• 2002
A new family of quadrangular or cubic mixed finite elements is constructed for the approximation of elastic wave equations and lead to explicit schemes, after time discretization, including in the case of anisotropic media. Expand
The Effect of Quadrature Errors on Finite Element Approximations for Second Order Hyperbolic Equations
• Mathematics
• 1976
The effects of numerical quadrature errors on finite element approximations to the solution of the mixed initial-boundary value problem for second order linear hyperbolic equations are studied. It isExpand
Construction and Analysis of Fourth-Order Finite Difference Schemes for the Acoustic Wave Equation in Nonhomogeneous Media
• Mathematics
• 1996
In this article, we construct and analyse a family of finite difference schemes for the acoustic wave equation with variable coefficients. These schemes are fourth-order accurate in space and time inExpand
A comparison between higher-order finite elements and finite differences for solving the wave equation
High-order finite elements with mass lumping allow for explicit time stepping when integrating the wave equation. An earlier study suggests that this approach can be used for two-dimensionalExpand
Conforming and nonconforming finite element methods for solving the stationary Stokes equations I
• Mathematics
• 1973
— The paper is devoted to a gênerai finite element approximation ofthe solution of the Stokes équations for an incompressible viscous fluid, Both conforming and nonconforming finite element methodsExpand
Spectral element methods for the incompressible Navier-Stokes equations
• Mathematics
• 1989
Spectral element methods are high-order weighted-residual techniques for partial differential equations that combine the geometric flexibility of finite element techniques with the rapid convergenceExpand
Error Estimates for Finite Element Methods for Second Order Hyperbolic Equations
The standard Galerkin method for a mixed initial-boundary value problem for a linear second order hyperbolic equation is analysed.Optimal estimates for the error in $L^\infty (L^2 )$ are derived us...
Spectral element method for acoustic wave simulation in heterogeneous media
• Physics
• 1994
Abstract In this paper, we present a spectral element method for studying acoustic wave propagation in complex geological structures. Due to complexity (both lithological and stratigraphical), theExpand
ACCURACY OF FINITE‐DIFFERENCE MODELING OF THE ACOUSTIC WAVE EQUATION
• Mathematics
• 1974
two methods rapidly deteriorates. This effect, known as “grid dispersion,” must be taken into account in order to avoid erroneous interpretation of seismograms obtained by finite-differenceExpand