# Convergence Analysis of a Discontinuous Galerkin Method with Plane Waves and Lagrange Multipliers for the Solution of Helmholtz Problems

@article{Amara2009ConvergenceAO, title={Convergence Analysis of a Discontinuous Galerkin Method with Plane Waves and Lagrange Multipliers for the Solution of Helmholtz Problems}, author={Mohamed Amara and Rabia Djellouli and Charbel Farhat}, journal={SIAM J. Numer. Anal.}, year={2009}, volume={47}, pages={1038-1066} }

We analyze the convergence of a discontinuous Galerkin method (DGM) with plane waves and Lagrange multipliers that was recently proposed by Farhat, Harari, and Hetmaniuk [Comput. Methods Appl. Mech. Engrg., 192 (2003), pp. 1389-1419] for solving two-dimensional Helmholtz problems at relatively high wavenumbers. We prove that the underlying hybrid variational formulation is well-posed. We also present various a priori error estimates that establish the convergence and order of accuracy of the…

## 52 Citations

### Plane Wave Discontinuous Galerkin Methods for the 2D Helmholtz Equation: Analysis of the p-Version

- Computer Science, MathematicsSIAM J. Numer. Anal.
- 2011

A priori convergence analysis of PWDG in the case of $p$-refinement is concerned, that is, the study of the asymptotic behavior of relevant error norms as the number of plane wave directions in the local trial spaces is increased.

### Dispersion analysis of plane wave discontinuous Galerkin methods

- Mathematics
- 2014

The plane wave DG (PWDG) method for the Helmholtz equation was introduced and analyzed in [GITTELSON, C., HIPTMAIR, R., AND PERUGIA, I. Plane wave discontinuous Galerkin methods: analysis of the…

### Hybridizable Discontinuous Galerkin Methods for Helmholtz Equation with High Wave Number. Part I: Linear case

- MathematicsArXiv
- 2020

This paper addresses several aspects of the linear Hybridizable Discontinuous Galerkin Method (HDG) for the Helmholtz equation with impedance boundary condition at high frequency. First, error…

### A STABILIZED DISCONTINUOUS GALERKIN FORMULATION FOR HELMHOLTZ PROBLEMS

- Computer Science
- 2008

A new discontinuous Galerkin formulation, based on a local approximation of the solution by plane waves that satisfy the wave equation, is proposed, built in a variational formulation framework that leads to a linear system associated with a positive definite Hermitian matrix.

### A Hybridizable Discontinuous Galerkin Method for the Helmholtz Equation with High Wave Number

- Computer ScienceSIAM J. Numer. Anal.
- 2013

Through choosing a specific parameter and using the duality argument, it is proved that the HDG method is stable without any mesh constraint for any wave number $\kappa$.

### Numerical methods for solving Helmholtz problems

- Mathematics, Computer Science
- 2009

The obtained numerical results clearly indicate that the proposed solution methodology outperforms standard finite element methods, as well as existing DG methodologies, such as the method proposed by Farhat et al (2003).

### Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

- Computer Science, MathematicsMath. Comput.
- 2013

To the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia] is extended, and convergence rates are derived in the particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions.

### Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the Helmholtz equation

- Mathematics, Computer Science
- 2013

An adaptive Interior Penalty Discontinuous Galerkin (IPDG) method based on adaptively refined simplicial triangulations of the computational domain yields convergence of the adaptive IPDG approach to the Helmholtz equation.

### A high-order discontinuous Galerkin method with Lagrange multipliers for advection-diffusion problems

- Mathematics, Computer Science
- 2013

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