# 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} }

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

#### 195 Citations

HIGHER-ORDER MASS-LUMPED FINITE ELEMENTS FOR THE WAVE EQUATION

- Mathematics
- 2001

The finite-element method (FEM) with mass lumping is an efficient scheme for modeling seismic wave propagation in the subsurface, especially in the presence of sharp velocity contrasts and rough… Expand

Error Estimates for Finite Element Approximations of a Viscous Wave Equation

- Mathematics
- 2011

We consider a family of fully discrete finite element schemes for solving a viscous wave equation, where the time integration is based on the Newmark method. A rigorous stability analysis based on… Expand

Mixed Higher Order Spectral Finite Elements for Reissner-Mindlin Equations

- Mathematics, Computer Science
- SIAM J. Sci. Comput.
- 2007

A class of high order numerical approximations for the Reissner-Mindlin plate model in the time domain, based on mixed spectral finite elements with mass lumping is constructed, which shows the advantages of the schemes in terms of accuracy and computational time. Expand

Boundary absorbing computations in wave simulation with the finite element method

- Mathematics
- 2011 Second International Conference on Mechanic Automation and Control Engineering
- 2011

The simulation of wave propagation has important applications in many problems such as in computational seismology. Here, we focus on boundary absorbing computations in wave simulation with the… Expand

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

- Computer Science, Mathematics
- Int. J. Appl. Math. Comput. Sci.
- 2007

Lagrange finite elements of order k are considered and it is shown how it is possible to automatically generate the mass and stiffness matrices of any order with the help of symbolic computation software. Expand

A comparison of continuous mass‐lumped finite elements with finite differences for 3‐D wave propagation

- Physics
- 2014

The finite-difference method on rectangular meshes is widely used for time-domain modelling of the wave equation. It is relatively easy to implement high-order spatial discretization schemes and… Expand

Foundations of Finite Element Methods for Wave Equations of Maxwell Type

- Mathematics, Computer Science
- Applied Wave Mathematics
- 2009

This paper treats convergence theory for linear second order evolution equations and includes studies of consistency and eigenvalue approximation, and emphasizes differential operators, such as the curl, which have large kernels, and use L2-stable interpolators preserving them. Expand

Different Types of Finite Elements

- Physics
- 2008

Our contribution concentrates on the numerical solution of acoustic wave problems applying the Finite Element Method (FEM). After a short introduction, we provide a detailed discussion about the… Expand

Numerical solution of the acoustic wave equation using Raviart-Thomas elements

- Mathematics
- 2007

In this paper we discuss the numerical approximation of the displacement form of the acoustic wave equation using mixed finite elements. The mixed formulation allows for approximation of both… Expand

Dispersion Properties of Explicit Finite Element Methods for Wave Propagation Modelling on Tetrahedral Meshes

- Computer Science, Mathematics
- J. Sci. Comput.
- 2018

The dispersion properties of two types of explicit finite element methods for modelling acoustic and elastic wave propagation on tetrahedral meshes are analyzed to give an indication of which method is the most efficient for a given accuracy, how many elements per wavelength are required for agiven accuracy, and how sensitive the accuracy of the method is to poorly shaped elements. Expand

#### 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 finite… Expand

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 is… Expand

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 in… Expand

A comparison between higher-order finite elements and finite differences for solving the wave equation

- Mathematics
- 1996

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-dimensional… Expand

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 methods… Expand

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 convergence… Expand

Error Estimates for Finite Element Methods for Second Order Hyperbolic Equations

- Mathematics
- 1976

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), the… Expand

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-difference… Expand