A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media

This paper develops and analyzes some interior penalty discontinuous Galerkin (IPDG) methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence well-posed) without any mesh constraint. For each fixed wave number $k$, optimal order (with respect to $h$) error estimate in the broken $H^1$-norm and suboptimal order estimate in the $L^2$-norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size $h$ is restricted to the preasymptotic regime (i.e., $k^2 h \gtrsim 1$). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to $k$) in the error bounds. The novelties of the proposed IPDG methods include the following: First, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts, so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a non-Hermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To this end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [P. Cummings, Analysis of Finite Element Based Numerical Methods for Acoustic Waves, Elastic Waves and Fluid-Solid Interactions in the Frequency Domain, Ph.D. thesis, The University of Tennessee, Knoxville, TN, 2001], [P. Cummings and X. Feng, Math. Models Methods Appl. Sci., 16 (2006), pp. 139-160], [U. Hetmaniuk, Commun. Math. Sci., 5 (2007), pp. 665-678].

Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

SUMMARY Recently, there has been an increased interest in applying the discontinuous Galerkin method (DGM) to wave propagation. In this work, we investigate the applicability of the interior penalty DGM to elastic wave propagation by analysing it’s grid dispersion properties, with particular attention to the effect that different basis functions have on the numerical dispersion. We consider different types of basis functions that naturally yield a diagonal mass matrix. This is relevant to seismology because a diagonal mass matrix is tantamount to an explicit and efficient time marching scheme. We find that the Legendre basis functions that are traditionally used in the DGM introduce numerical dispersion and anisotropy. Furthermore, we find that using Lagrange basis functions along with the Gauss nodes has attractive advantages for numerical wave propagation.

https://www.math.purdue.edu/~santos/research/disc_galerkin_ref/DeBasabe-etal-2008-GJI.pdf

The interior penalty discontinuous Galerkin method for elastic wave propagation: grid dispersion

High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics

In this paper, we develop and analyze a new class of discontinuous Galerkin (DG) methods for the acoustic wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches are typically dissipative or suboptimally convergent, depending on the choice of numerical fluxes. Our new method can be seen as a compromise between these two kinds of techniques, in the way that it is both explicit and energy conserving, locally and globally. Moreover, it can be seen as a generalized version of the Raviart-Thomas FE method and the finite volume method. Stability and convergence of the new method are rigorously analyzed, and we have shown that the method is optimally convergent. Furthermore, in order to apply the new method for unbounded domains, we apply our new method with the first order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed.

/pdf/optimal-discontinuous-galerkin-methods-for-the-acoustic-wave-1e5wyow2d3.pdf

Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions

We have developed and analyzed a new class of discontinuous Galerkin methods (DG) which can be seen as a compromise between standard DG and the finite element (FE) method in the way that it is explicit like standard DG and energy conserving like FE. In the literature there are many methods that achieve some of the goals of explicit time marching, unstructured grid, energy conservation, and optimal higher order accuracy, but as far as we know only our new algorithms satisfy all the conditions. We propose a new stability requirement for our DG. The stability analysis is based on the careful selection of the two FE spaces which verify the new stability condition. The convergence rate is optimal with respect to the order of the polynomials in the FE spaces. Moreover, the convergence is described by a series of numerical experiments.

/pdf/optimal-discontinuous-galerkin-methods-for-wave-propagation-4cdijj5qhd.pdf

Bjo¨rn Engquist

Papers

Optimal Discontinuous Galerkin Methods for Wave Propagation