English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we develop and analyze some interior penalty hp- discontinuous Galerkin (hp-DG) methods for the Helmholtz equation with first order absorbing boundary condition in two and three dimensions. The pro- posed hp-DG methods are defined using a sesquilinear form which is not only mesh-dependent (or h-dependent) but also degree-dependent (or p-dependent). In addition, the sesquilinear form contains penalty terms which not only pe- nalize the jumps of the function values across the element edges but also the jumps of the first order tangential derivatives as well as jumps of all normal derivatives up to order p. Furthermore, to ensure the stability, the penalty parameters are taken as complex numbers with positive imaginary parts, so essentially and practically no constraint is imposed on the penalty parameters. It is proved that the proposed hp-discontinuous Galerkin methods are stable (hence, well-posed) without any mesh constraint. For each fixed wave number k, sub-optimal order (with respect to h and p) error estimates in the broken H1-norm and the L2-norm are derived without any mesh constraint. The er- ror estimates as well as the stability estimates are improved to optimal order under the mesh condition k3h2p-2 ≤ C0 by utilizing these stability and error estimates and using a stability-error iterative procedure, where C0 is some constant independent of k, h, p, and the penalty parameters. To overcome the difficulty caused by strong indefiniteness (and non-Hermitian nature) of the Helmholtz problems in the stability analysis for numerical solutions, our main ideas for stability analysis are 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 [19,20, 33], which enable us to derive stability estimates and error bounds with explicit dependence on the mesh size h, the polynomial degree p, thewavenumberk, aswellasallthepenaltyparametersforthe numerical solutions. © 2011 American Mathematical Society.

ℎ𝑝-Discontinuous Galerkin methods for the Helmholtz equation with large wave number