Stability and convergence of fully discrete finite element schemes for the acoustic wave equation

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4 Citations (Scopus)

Abstract

In this paper, we investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. A convergence analysis is carried out and optimal a priori error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L2-norm error over a finite time interval converges optimally as O(hp+1+Δts ), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and Δt the time step.

Original languageEnglish
Pages (from-to)659-682
Number of pages24
JournalJournal of Applied Mathematics and Computing
Volume40
Issue number1-2
DOIs
Publication statusPublished - Oct 2012

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Stability and Convergence
Acoustic Waves
Wave equations
Wave equation
Acoustic waves
Finite Element
A Priori Error Estimates
Optimal Error Estimates
Discontinuous Galerkin
Energy Method
Time Stepping
Smooth Solution
Convergence Analysis
Stability Analysis
Covering
Discretization
Mesh
Denote
Converge
Norm

Keywords

  • Discontinuous Galerkin method
  • Energy method
  • Newmark scheme
  • Optimal error estimates
  • Stability condition
  • Wave equation

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics

Cite this

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abstract = "In this paper, we investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. A convergence analysis is carried out and optimal a priori error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L2-norm error over a finite time interval converges optimally as O(hp+1+Δts ), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and Δt the time step.",
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AB - In this paper, we investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. A convergence analysis is carried out and optimal a priori error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L2-norm error over a finite time interval converges optimally as O(hp+1+Δts ), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and Δt the time step.

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