Comparison of Least Squares FEM, Mixed Galerkin FEM and an Implicit FDM Applied to Acoustic Scattering

Abstract

This paper compares the performances of a least squares finite element method (LSFEM), a mixed Galerkin finite element method (MGFEM), and an implicit finite difference method (IFDM), when they are applied to an acoustic scattering problem modelled by the one-dimensional Helmholtz equation. First, the boundary value problem is written as a first order system of differential equations, and variational formulations for the LSFEM and MGFEM are derived. Then, by using appropriate basis functions, the stiffness matrices and load vectors which define the corresponding linear algebraic systems are obtained. In all cases, the stiffness matrix is banded and for the LSFEM it is also Hermitian. Numerical tests show that all these methods have quadratic convergence to the exact solution. However, their efficiency assessed in terms of the number of nodes and computing time required to reach quadratic convergence varies. It is observed that LSFEM requires many more nodes and employs much more time than MGFEM and IFDM to reach the quadratic convergence for high frequencies. It is also found that MGFEM has less numerical dispersion and, as a consequence, performs better than IFDM and LSFEM for high frequencies. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)