Dispersion free analysis of acoustic problems using the alpha finite element method

Abstract

The classical finite element method (FEM) fails to provide accurate results to the Helmholtz equation with large wave numbers due to the well-known “pollution error” caused by the numerical dispersion, i.e. the numerical wave number is always smaller than the exact one. This dispersion error is essentially rooted at the “overly-stiff” feature of the FEM model. In this paper, an alpha finite element method (α-FEM) is then formulated for the acoustic problems by combining the “smaller wave number” model of FEM and the “larger wave number” model of NS-FEM through a scaling factor \({a\in [0,1]}\) . The motivation for this combined approach is essentially from the features of “overly-stiff” FEM model and “overly-soft” NS-FEM model, and accurate solutions can be obtained by tuning the α-FEM model. A technique is proposed to determine a particular alpha with which the α-FEM model can possess a very “close-to-exact” stiffness, which can effectively reduce the dispersion error leading to dispersion free solutions for acoustic problems. Theoretical and numerical studies shall demonstrate the excellent properties of the present α-FEM.