Explicit Discrete Dispersion Relations for the Acoustic Wave Equation in d-Dimensions Using Finite Element, Spectral Element and Optimally Blended Schemes

This work concerns the error analysis of the spectral element method with Gauss–Lobatto–Chebyshev collocation points with the implicit Newmark average acceleration scheme for the two-dimensional acoustic wave equation. The analysis is restricted to homogeneous Dirichlet boundary conditions, constant compressibility and variable density. The proposed error estimates are optimal with respect to the mesh parameter although suboptimal on the polynomial degree. Numerical examples illustrate the theoretical results.

Download Full-text

ADAPTIVE FINITE ELEMENT TECHNIQUES FOR THE ACOUSTIC WAVE EQUATION

Journal of Computational Acoustics ◽

10.1142/s0218396x01000668 ◽

2001 ◽

Vol 09 (02) ◽

pp. 575-591 ◽

Cited By ~ 29

Author(s):

WOLFGANG BANGERTH ◽

ROLF RANNACHER

Keyword(s):

Finite Element ◽

Wave Equation ◽

Acoustic Wave ◽

Adaptive Finite Element Method ◽

Adaptive Finite Element ◽

Efficient Computation ◽

Acoustic Wave Equation ◽

Grid Adaptation ◽

Local Cell ◽

Error Indicators

We present an adaptive finite element method for solving the acoustic wave equation. Using a global duality argument and Galerkin orthogonality, we derive an identity for the error with respect to an arbitrary functional output of the solution. The error identity is evaluated by solving the dual problem numerically. The resulting local cell-wise error indicators are used in the grid adaptation process. In this way, the space-time mesh can be tailored for the efficient computation of the quantity of interest. We give an overview of the implementation of the proposed method and illustrate its performance by several numerical examples.

Download Full-text

Finite Element θ-Schemes for the Acoustic Wave Equation

Advances in Applied Mathematics and Mechanics ◽

10.4208/aamm.10-m1018 ◽

2011 ◽

Vol 3 (1) ◽

pp. 181-203 ◽

Cited By ~ 6

Author(s):

Samir Karaa

Keyword(s):

Finite Element ◽

Wave Equation ◽

Acoustic Wave ◽

Stability And Convergence ◽

Optimal Error Estimates ◽

Finite Difference Schemes ◽

Time Interval ◽

Acoustic Wave Equation ◽

Time Step ◽

Polynomial Degree

AbstractIn this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal 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 + ∆ts), where p denotes the polynomial degree, s=2 or 4, h the mesh size, and ∆t the time step.

Download Full-text