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

2012 ◽  
Vol 40 (1-2) ◽  
pp. 659-682 ◽  
Author(s):  
Samir Karaa
Keyword(s):  
Finite Element ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Stability And Convergence ◽  
Acoustic Wave Equation ◽  
Fully Discrete ◽  
Discrete Finite Element ◽  
Finite Element Schemes

Finite Element θ-Schemes for the Acoustic Wave Equation

2011 ◽  
Vol 3 (1) ◽  
pp. 181-203 ◽  
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.

Finite Element Analysis of Maxwell’s Equations in Dispersive Lossy Bi-Isotropic Media

2013 ◽  
Vol 5 (04) ◽  
pp. 494-509 ◽  
Author(s):  
Yunqing Huang ◽  
Jichun Li ◽  
Yanping Lin
Keyword(s):  
Finite Element ◽  
Existence And Uniqueness ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Practical Implementation ◽  
Element Analysis ◽  
Fully Discrete ◽  
Isotropic Media ◽  
Discrete Finite Element ◽  
Finite Element Schemes

AbstractIn this paper, the time-dependent Maxwell’s equations used to modeling wave propagation in dispersive lossy bi-isotropic media are investigated. Existence and uniqueness of the modeling equations are proved. Two fully discrete finite element schemes are proposed, and their practical implementation and stability are discussed.

ADAPTIVE FINITE ELEMENT TECHNIQUES FOR THE ACOUSTIC WAVE EQUATION

2001 ◽  
Vol 09 (02) ◽  
pp. 575-591 ◽  
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.

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