A Galerkin least‐squares finite element method for the two‐dimensional Helmholtz equation

Abstract

SUMMARYIn this paper a Galerkin least-squares (GLS) finite element method, in which residuals in least-squares form are added to the standard Galerkin variational equation, is developed to solve the Helmholtz equation in two dimensions. An important feature of GLS methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes. Previous work has accomplished this for the one-dimensional Helmholtz equation using dispersion analysis. In this paper, the selection of the GLS mesh parameter for two dimensions is considered, and leads to elements that exhibit improved phase accuracy. For any given direction of wave propagation, an optimal GLS mesh parameter is determined using two-dimensional Fourier analysis. In general problems, the direction of wave propagation will not be known a priori. In this case, an optimal GLS parameter is found which reduces phase error for all possible wave vector orientations over elements. The optimal GLS parameters are derived for both consistent and lumped mass approximations. Several numerical examples are given and the results compared with those obtained from the Galerkin method. The extension of GLS to higher-order quadratic interpolations is also presented.

Preconditioning Helmholtz linear systems

“…We discretize the system using an isoparametric discretization over quad elements as shown in figure 6. The analytic solution to this problem is known, and can be found in [5,26,24].…”

The Tailored Finite Point Method

“…In recent decades, many scientists presented e cient methods for this class of problems with constant coe cients, including the discrete singular convolution method [2], the hybrid numerical asymptotic method [13], the spectral approximation method [54], the element-free Galerkin method [59,63], the so-called ultra weak variational formulation [34], and the hybrid numerical-asymptotic boundary integral method [8]. In general, these methods need the restriction ℎ = O(1) for the mesh size ℎ in the simulation.…”

Comparison between results of solution of Burgers’ equation and Laplace’s equation by Galerkin and least-square finite element methods

“…Results of elements with four nodes were more accurate than results of other elements. Thompson and Pinsky (1995) utilized Galerkin least-squares finite element method (GLS) for solving the two-dimensional Helmholtz equation. This equation is a form of Laplace's equation and makes used of modeling of wave movement.…”