Discontinuous Galerkin methods for the acoustic vibration problem

TL;DR

This paper analyzes discontinuous Galerkin methods for the acoustic vibration problem, focusing on convergence, error estimates, and the influence of stabilization parameters on spectrum accuracy in 2D and 3D.

Contribution

It introduces and compares displacement and pressure formulations of DG methods for acoustic eigenvalue problems, providing convergence analysis and practical insights.

Findings

Convergence and error estimates are established for the methods.

The influence of stabilization parameters on spurious eigenvalues is characterized.

Pressure formulation offers a viable alternative with comparable performance.

Abstract

In two and three dimension we analyze discontinuous Galerkin methods for the acoustic problem. The acoustic fluid that we consider on this paper is inviscid, leading to a linear eigenvalue problem. The acoustic problem is written, in first place, in terms of the displacement. Under the approach of the non-compact operators theory, we prove convergence and error estimates for the method when the displacement formulation is considered. We analyze the influence of the stabilization parameter on the computation of the spectrum, where spurious eigenmodes arise when this parameter is not correctly chosen. Alternatively we present the formulation depending only on the pressure, comparing the performance of the DG methods with the pure displacement formulation. Computationally, we study the influence of the stabilization parameter on the arising of spurious eigenvalues when the spectrum is computed. Also, we report tests in two and three dimensions where convergence rates are reported, together with a comparison between the displacement and pressure formulations for the proposed DG methods.