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

Abstract

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. Abstract. We study the properties of a novel discontinuous Petrov Galerkin (DPG) method for acoustic wave propagation. The method yields Hermitian positive definite matrices and has good pre-asymptotic stability properties. Numerically, we find that the method exhibits negligible phase errors (otherwise known as pollution errors) even in the lowest order case. Theoretically, we are able to prove error estimates that explicitly show the dependencies with respect to the wavenumber ω, the mesh size h, and the polynomial degree p. But the current state of the theory does not fully explain the remarkably good numerical phase errors. Theoretically, comparisons are made with several other recent works that gave wave number explicit estimates. Numerically, comparisons are made with the standard finite element method and its recent modification for wave propagation with clever quadratures. The new DPG method is designed following the previously established principles of optimal test functions. In addition to the nonstandard test functions, in this work, we also use a nonstandard wave number dependent norm on both the test and trial space to obtain our error estimates.

Robust operator estimates and the application to substructuring methods for first-order systems

“…The efficiency of the preconditioning is demonstrated for diffusion and elasticity. Further numerical results for the Poisson problem [15], for the transport equation [13], for convection diffusion problems [12,18], for the Helmholtz problem [16], and for the Stokes problem [30] underline the flexibility and efficiency of discontinuous Petrov-Galerkin methods. Nevertheless, to exploit the full potential of a Petrov-Galerkin approach, further research is required to identify the optimal balance of local approximations and skeleton discretizations.…”

“…However, the choice of suitable finite dimensional test and trial spaces is far from trivial in the conforming setting since uniform infsup conditions have to be satisfied. Recently introduced discontinuous Petrov-Galerkin schemes [15,16,18,34] are of special interest and provide attractive alternatives to standard conforming schemes, see, also e.g., [9,13,17].…”

“…More precisely, we show that a broad class of linear partial differential equations can be efficiently approximated by methods which allow for a local reduction to positive definite Schur complement problems. Schur complement reductions are extensively investigated for finite element methods of least-squares type [4,5] and, recently, for least-square problems on element level resulting in discontinuous Petrov-Galerkin methods [15,16]. In combination with the reduction to the interface [10,25,27] efficient iterative solvers can be designed.…”

“…Proposition 8.1.4,[16,19], Section 4.3, the stability estimate can be shown in three steps. Firstly, we provide upper bounds for p ∂Ω and σ Ω .…”

Numerical Simulations of Cloaking Problems using a DPG Method

“…For reader's convenience, here we review quickly these main points in context of linear acoustics, see [23,8] for details.…”