Discontinuous Galerkin Methods for the Helmholtz Equation with Large Wave Number

Abstract

Abstract. This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the 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 sub-optimal 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 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 interior penalty discontinuous Galerkin methods include: 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 the 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 [23,24,35].

Is the Helmholtz Equation Really Sign-Indefinite?

“…Finally, recall that interior penalty methods arise by adding terms to the appropriate sesquilinear forms to penalize jumps of various quantities over interfaces between elements of a mesh. Although the variational formulations of these methods are not coercive, for certain methods some of the consequences of coercivity hold; namely, the Galerkin equations have a unique solution without any constraint on the dimension of the (piecewise-polynomial) approximation space, and error estimates can be obtained that are explicit in k, h, and p [37 [37] and [38], the penalty terms are added so that the properties just highlighted can be proved using Rellich-type identities. (These methods share a conceptual link with the new variational formulations introduced in this paper, since, as we see in section 1.4, the new formulations in this paper are designed using closely-related Morawetz-type identities.)…”

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

“…Examples of non-adaptive discretization are: standard finite differences [63,77], standard continuous or discontinuous finite elements [34,52,53,83,98], and spectral methods [79,100,101], among many others. They are very general in the sense that they can be used for a variety of different problems.…”

Wavenumber Explicit Analysis for a DPG Method for the Multidimensional Helmholtz Equation

“…We will compare our result with this work. There have been important recent developments in DG methods for the Helmholtz equation [12,13,16,23]. Ultraweak formulations, ever since the works of [4,17], have shown great potential in numerical solution of the Helmholtz equation, especially those approximations based on plane waves.…”