Discontinuous Galerkin Finite Element Method for the Wave Equation.
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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ABSTRACT: Purely numerical methods based on the Finite Element Method (FEM) are becoming increasingly popular in seismic modeling for the propagation of acoustic and elastic waves in geophysical models. These methods o er a better control on the accuracy and more geometrical exibility than the Finite Di erence methods that have been traditionally used for the generation of synthetic seismograms. However, the success of these methods has outpaced their analytic validation. The accuracy of the FEMs used for seismic wave propagation is unknown in most cases and therefore the simulation parameters in numerical experiments are determined by empirical rules. I focus on two methods that are particularly suited for seismic modeling: the Spectral Element Method (SEM) and the Interior-Penalty Discontinuous Galerkin Method (IP-DGM). The goals of this research are to investigate the grid dispersion and stability of SEM and IP-DGM, to implement these methods and to apply them to subsurface models to obtain synthetic seismograms. In order to analyze the grid dispersion and stability, I use the von Neumann method (plane wave analysis) to obtain a generalized eigenvalue problem. I show that the eigenvalues are related to the grid dispersion and that, with certain assumptions, the size of the eigenvalue problem can be reduced from the total number of degrees of freedom to one proportional to the number of degrees of freedom inside one element. The grid dispersion results indicate that SEM of degree greater than 4 is isotropic and has a very low dispersion. Similar dispersion properties are observed for the symmetric formulation of IP-DGM of degree greater than 4 using nodal basis functions. The low dispersion of these methods allows for a sampling ratio of 4 nodes per wavelength to be used. On the other hand, the stability analysis shows that, in the elastic case, the size of the time step required in IP-DGM is approximately 6 times smaller than that of SEM. The results from the analysis are con rmed by numerical experiments performed using an implementation of these methods. The methods are tested using two benchmarks: Lamb's problems and the SEG/EAGE salt dome model. Computational Science, Engineering, and Mathematics Program
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ABSTRACT: Finite difference operators approximating second derivatives with variable coefficients and satisfying a summation-by-parts rule have been derived for the second-, fourth- and sixth-order case by using the symbolic mathematics software Maple. The operators are based on the same norms as the corresponding approximations of the first derivative, which makes the construction of stable approximations to general multi-dimensional hyperbolic-parabolic problems straightforward. KeywordsHigh-order finite difference methods–Numerical stability–Second-derivatives–Variable coefficientsJournal of Scientific Computing 06/2011; 51(3):650-682. · 1.71 Impact Factor
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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.Journal of Applied Mathematics and Computing 01/2012; 40(1-2).