Article

# Discontinuous Galerkin Finite Element Method for the Wave Equation

(Impact Factor: 1.79). 01/2006; 44(6):2408-2431. DOI: 10.1137/05063194X
Source: DBLP

ABSTRACT

The symmetric interior penalty discontinuous Galerkin nite element method is presented for the numerical discretization of the second-order scalar wave equation. The resulting stiness matrix is symmetric positive denite and the mass matrix is essentially diagonal; hence, the method is inherently parallel and leads to fully explicit time integration. Optimal a priori error bounds are derived in the energy norm and the L2-norm for the semi-discrete formulation. In particular, the error in the energy norm is shown to converge with the optimal orderO(hminfs;'g) with respect to the mesh size h, the polynomial degree ', and the regularity exponent s of the continuous solution. Under additional regularity assumptions, the L2-error is shown to converge with the optimal orderO(h'+1). Numerical results results conrm the expected convergence rates and illustrate the versatility of the method.

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Available from: Marcus J. Grote, Feb 01, 2015
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• "For a general introduction to the subject, see Cockburn et al. (2000); Riviere (2008); Arnold (1982). Regarding computational wave propagation, some discontinuous Galerkin methods have been proposed in Bernacki et al. (2006); Bourdel et al. (1991); Giraldo et al. (2002); Grote et al. (2006); Hu et al. (1999) for the acoustic wave equations and in Falk and Richter (1999); Johnson and Pitkäranta (1986) for the hyperbolic system. For seismic wave simulations, some discontinuous Galerkin methods have been proposed in De Basabe et al. (2008); Dumbser and Kaser (2006);Rivì ere and Wheeler (2003). "
##### Article: A staggered discontinuous Galerkin method for the simulation of seismic waves with surface topography
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ABSTRACT: Accurate simulation of seismic waves is of critical importance in a variety of geophysical applications. Based on recent works on staggered discontinuous Galerkin methods, we have developed a new method for the simulations of seismic waves, which has energy conservation and extremely low grid dispersion, so that it naturally provided accurate numerical simulations of wave propagation useful for geophysical applications and was a generalization of classical staggered-grid finite-difference methods. Moreover, it could handle with ease irregular surface topography and discontinuities in the subsurface models. Our new method discretized the velocity and the stress tensor on this staggered grid, with continuity imposed on different parts of the mesh. The symmetry of the stress tensor was enforced by the Lagrange multiplier technique. The resulting method was an explicit scheme, requiring the solutions of a block diagonal system and a local saddle point system in each time step, and it was, therefore, very efficient. To tailor our scheme to Rayleigh waves, we developed a mortar formulation of our method. Specifically, a fine mesh was used near the free surface and a coarse mesh was used in the rest of the domain. The two meshes were in general not matching, and the continuity of the velocity at the interface was enforced by a Lagrange multiplier. The resulting method was also efficient in time marching. We also developed a stability analysis of the scheme and an explicit bound for the time step size. In addition, we evaluated some numerical results and found that our method was able to preserve the wave energy and accurately computed the Rayleigh waves. Moreover, the mortar formulation gave a significant speed up compared with the use of a uniform fine mesh, and provided an efficient tool for the simulation of Rayleigh waves.
Geophysics 05/2015; 80(4):T119-T135. DOI:10.1190/geo2014-0413.1 · 1.61 Impact Factor
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• "A summary of SBP-SAT methodology is made in [3]. Another candidate for the approximation of spatial derivatives is discontinuous Galerkin (dG) method, and it has been successfully used in [4] for the wave equation. dG can handle complex geometry of the computational domain, but often has more degrees of freedom than finite difference method for the same accuracy. "
##### Article: Convergence analysis for the SBP-SAT approximation of the second order wave equation
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ABSTRACT: In this paper, we consider the second order wave equation discretized in space by summation-by-parts-simultaneous approximation term (SBP-SAT) technique. Special emphasis is placed on the accuracy analysis of the treatment of the Dirichlet boundary condition and of the grid interface condition. The result shows that a boundary or grid interface closure with truncation error $\mathcal{O}(h^p)$ converges of order $p + 2$ if the penalty parameters are chosen carefully. We show that stability does not automatically yield a gain of two orders in convergence rate. The accuracy analysis is verified by numerical experiments.
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• "Even though explicit numerical time integration may be applied directly to (2.3), every time-step then requires the solution of a linear system involving M. To avoid that computational work, various mass-lumping techniques have been developed [12] [35], which replace M by a diagonal approximation without spoiling the accuracy [4]. Alternatively, the spectral element method [8] [31] and the symmetric interior penalty DG method [21] both waive the need for mass-lumping altogether: The former inherently leads to a diagonal mass matrix, whereas the latter leads to a block-diagonal mass matrix with block size equal to the number of degrees of freedom per element. Thus, both alternative FE discretizations also lead to (2.3) with an essentially diagonal mass matrix M. "
##### Article: Runge--Kutta-Based Explicit Local Time-Stepping Methods for Wave Propagation
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ABSTRACT: Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for the simulation of time-dependent wave phenomena. By taking smaller time-steps precisely where the smallest elements are located, local time-stepping (LTS) methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh. Starting from classical or low-storage explicit RK methods, explicit LTS methods of arbitrarily high accuracy are derived. When combined with an essentially diagonal finite element mass matrix, the resulting time-marching schemes retain the high accuracy, stability, and efficiency of the original RK methods while circumventing the geometry-induced stiffness. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these LTS-RK methods.
SIAM Journal on Scientific Computing 03/2015; 37(2):2015. DOI:10.1137/140958293 · 1.85 Impact Factor