On the Performance of a Stabilized Discontinuous Galerkin-Type Method for Solving Efficiently Helmholtz Problems

Abstract eng:
The Helmholtz equation, that describes time-harmonic wave propagation, belongs to the classical equations of mathematical physics that are well understood from a mathematical view point. However, computing the solutions for large wavenumbers is still a challenging problem in spite the tremendous progress made in the last two decades. We propose a solution methodology that falls in the category of discontinuous Galerkin methods for the solution of mid- to high-frequency Helmholtz problems. The method can be viewed “between” the Discontinuous Galerkin method [1,2] and the Least-Squares method [3]. The proposed method is based on a decomposition of the domain in quadrilateral- or triangular-shaped elements. The solution is approximated, at the element level, by a superposition of plane waves that are solution of the Helmholtz equation. The continuity of the solution and of its normal derivative at the interior interfaces of the elements is then enforced in the least-squares sense using Lagrange multipliers. Note that the proposed technique is a two-step procedure, where first positive definite linear systems are solved in parallel at the element level and then, the Lagrange multipliers are evaluated by solving a linear system with positive semi-definite matrix. The numerical results obtained in the case of waveguide problems clearly demonstrate the stability of the method as well as its high level of accuracy. For example, for the so-called R-7-2 element, the method remains stable for a mesh resolution with over 1000 elements per wavelength. In addition, the proposed method exhibits an impressive level of accuracy in the high frequency regime. Indeed, for ka = 400, where k is the wavenumber and a characterizes the dimension of the domain, and with about 3 elements per wavelength only, the proposed solution methodology equipped with the R-11-3 element delivers a total relative error of less than 0.6%. References [1] C. Farhat, I. Harari, U. Hetmaniuk, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime, Comput. Methods Appli. Mech. Eng., 192, 1389-1419, 2003. [2] C. Farhat, P. Wiedemann-Goiran, R. Tezaur, A discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of short wave exterior Helmholtz problems on unstructured meshes, Wave Motion, 39, 307-317, 2004. [3] P. Monk, D.Q. Wang, A least-squares method for the Helmholtz equation, Comput. Methods Appli. Mech. Eng., 175, 411-454, 1999.

• Export as: BibTeX | MARC | MARCXML | DC | EndNote | NLM | RefWorks
• Add to your basket