Discontinuous Galerkin Finite Element Method for the Wave Equation

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


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). "
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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. "
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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. "
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