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 language | English |
|---|---|
| Pages (from-to) | 659-682 |
| Number of pages | 24 |
| Journal | Journal of Applied Mathematics and Computing |
| Volume | 40 |
| Issue number | 1-2 |
| DOIs | |
| Publication status | Published - Oct 2012 |
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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
Stability and convergence of fully discrete finite element schemes for the acoustic wave equation. / Karaa, Samir.
In: Journal of Applied Mathematics and Computing, Vol. 40, No. 1-2, 10.2012, p. 659-682.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Stability and convergence of fully discrete finite element schemes for the acoustic wave equation
AU - Karaa, Samir
PY - 2012/10
Y1 - 2012/10
N2 - 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.
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.
KW - Discontinuous Galerkin method
KW - Energy method
KW - Newmark scheme
KW - Optimal error estimates
KW - Stability condition
KW - Wave equation
UR - http://www.scopus.com/inward/record.url?scp=84865726575&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84865726575&partnerID=8YFLogxK
U2 - 10.1007/s12190-012-0558-8
DO - 10.1007/s12190-012-0558-8
M3 - Article
AN - SCOPUS:84865726575
VL - 40
SP - 659
EP - 682
JO - Journal of Applied Mathematics and Computing
JF - Journal of Applied Mathematics and Computing
SN - 1598-5865
IS - 1-2
ER -