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Journal of Applied Mathematics and Computing

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01-10-2012 | Computational mathematics

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

Author: Samir Karaa

Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2012

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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 L ²-norm error over a finite time interval converges optimally as O(h ^p+1+Δt ^s), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and Δt the time step.

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Title: Stability and convergence of fully discrete finite element schemes for the acoustic wave equation
Author: Samir Karaa
Publication date: 01-10-2012
Publisher: Springer-Verlag
Published in: Journal of Applied Mathematics and Computing / Issue 1-2/2012
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-012-0558-8

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