TY - JOUR
T1 - Stability and convergence of fully discrete finite element schemes for the acoustic wave equation
AU - Karaa, Samir
N1 - Funding Information:
This research was supported by Sultan Qaboos University under Grant IG/SCI/DOMS/11/09.
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
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U2 - 10.1007/s12190-012-0558-8
DO - 10.1007/s12190-012-0558-8
M3 - Article
AN - SCOPUS:84865726575
SN - 1598-5865
VL - 40
SP - 659
EP - 682
JO - Journal of Applied Mathematics and Computing
JF - Journal of Applied Mathematics and Computing
IS - 1-2
ER -